PERTURBATIONS OF ORTHOGONAL POLYNOMIALS WITH PERIODIC RECURSION COEFFICIENTS

DAVID DAMANIK^{1}, ROWAN KILLIP^{2}, AND BARRY SIMON^{3}

Abstract. We extend the results of Denisov–Rakhmanov, Szegő–Shohat– Nevai, and Killip–Simon from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC. The key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations.

1. Introduction

This is a paper about the spectral theory of orthogonal polynomials on the real line (OPRL) and orthogonal polynomials on the unit circle (OPUC), that is, the connection of the underlying (spectral) measure and the recursion coefficients.

Specifically, given a probability measure, dρ, on ℝ with bounded but infinite support, the orthonormal polynomials, p_{n}(x), obey a recursion relation

xp_{n}(x) = a_{n+1}p_{n+1}(x) + b_{n+1}p_{n}(x) + a_{n}p_{n−1}(x)  (1.1)

where the Jacobi parameters {a_{n}, b_{n}}_{n=1}^{∞} obey b_{j} ∈ ℝ, a_{j} ∈ (0, ∞). As is well known (see, e.g., [98, Section 1.3]), (1.1) sets up a one-one correspondence between uniformly bounded {a_{n}, b_{n}}_{n=1}^{∞} and such measures, dρ (this is sometimes called Favard’s theorem).

Similarly, probability measures, dμ, on ∂𝔻 which are nontrivial (i.e., their support is not a finite set of points) are in one-one correspondence with sequences {α_{n}}_{n=0}^{∞} of Verblunsky coefficients in 𝔻 ≡ {z : |z| < 1} via the recursion relation of the orthonormal polynomials φ_{n}(z), namely,

zφ_{n}(z) = ρ_{n}φ_{n+1}(z) + ᾱ_{n}φ_{n}^{∗}(z)  (1.2)

where

φ_{n}^{∗}(z) = z^{n} \overline{φ_{n}(1/z̄)}  ρ_{n} = (1 − |α_{n}|^{2})^{1/2}  (1.3)

Underlying the association of measures and recursion coefficients are matrix representations. For OPRL, we take the matrix for multiplication by x in the {p_{n}}_{n=0}^{∞} basis of L^{2}(ℝ, dρ), which is the tridiagonal Jacobi matrix

Date: February 2, 2007.

1 Department of Mathematics, Rice University, Houston, TX 77005. E-mail:

damanik@rice.edu. Supported in part by NSF grant DMS-0500910.

2 Department of Mathematics, UCLA, Los Angeles, CA 90095. E-mail: killip@math.ucla.edu.

Supported in part by NSF grant DMS-0401277 and a Sloan Foundation Fellowship.

3 Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125. E-mail: bsimon@caltech.edu. Supported in part by NSF grant DMS-0140592 and U.S.–Israel Binational Science Foundation (BSF) Grant No. 2002068.

J = (\begin{matrix}b_{1} & a_{1} & 0 \\ a_{1} & b_{2} & a_{2} & ... \\ 0 & a_{2} & ... & ... \\ & ... & ... & ... \end{matrix})  (1.4)

For OPUC, one takes the matrix, C, for multiplication by z in the basis obtained by orthonormalizing {1, z, z^{−1}, z^{2}, z^{−2}, … } in L^{2}(∂𝔻, dμ). This CMV matrix (see [98, Section 4.2]) has the form

C = LM  (1.5)
L = Θ(α_{0}) ⊕ Θ(α_{2}) ⊕ ⋯  (1.6)
M = 1_{1×1} ⊕ Θ(α_{1}) ⊕ Θ(α_{3}) ⊕ ⋯  (1.7)
Θ(α) = (\begin{matrix}ᾱ & ρ \\ ρ & −α \end{matrix})  (1.8)

Note that C is unitary, while J is self-adjoint.

As a model for what we wish to prove, let us briefly survey some of the main results relating to (slowly decaying) perturbations of the free case, that is, a_{n} ≡ 1, b_{n} ≡ 0 for OPRL and α_{n} ≡ 0 for OPUC.

(1) Weyl’s Theorem [111, 13, 1, 43, 49]. If a_{n} → 1, b_{n} → 0, then σ_{ess}(dρ) = [−2, 2] and if α_{n} → 0, then σ_{ess}(dμ) = ∂𝔻. Here σ_{ess}(dρ) is the support of the measure, dρ, with all isolated points removed.

(2) Denisov–Rakhmanov Theorem [86, 87, 74, 30, 82]. If σ_{ess}(dρ) = Σ_{ac}(dρ) = [−2, 2], then a_{n} → 1 and b_{n} → 0. If σ_{ess}(dμ) = Σ_{ac}(dμ) = ∂𝔻, then α_{n} → 0. Here Σ_{ac}(dρ) is defined by: If dρ = f(x) dx + dρ_{s} (in the OPUC case, dx is dθ/2π) with dρ_{s} singular, then Σ_{ac}(dρ) = {x : f(x) ≠ 0} as a measure class, that is, a set modulo sets of Lebesgue measure zero.

(3) Szegő’s Theorem [105, 106, 94, 79, 59]. In the OPUC case, define

Z(dμ) ≡ − ∫ log(\frac{dμ_{ac}}{dθ} ) \frac{dθ}{2π}  (1.9)

Then Z(dμ) < ∞ if and only if

∑_{j=0}^{∞} |α_{j}(dμ)|^{2} < ∞  (1.10)

In the OPRL case, define

Z(dρ) ≡ − ∫_{−2}^{2} log(2π(4 − E^{2})^{1/2} \frac{dρ_{ac}}{dE} ) \frac{dE}{2π(4 − E^{2})^{1/2}}  (1.11)

Then, if we assume supp(dρ) ⊂ [−2, 2], we have

Z(dρ) < ∞ ⟺ lim sup ∑_{j=1}^{N} log(a_{j}) > −∞  (1.12)

and if that holds, then

∑_{n=1}^{∞}(a_{n} − 1)^{2} + b_{n}^{2} < ∞  (1.13)

and

∑(a_{n} − 1) and ∑ b_{n}  (1.14)

are conditionally convergent to finite numbers.

(4) Killip–Simon Theorem [59]. For OPRL, define

Q(dρ) = − ∫_{−2}^{2} log(π(4 − E^{2})^{−1/2} \frac{dρ_{ac}}{dE} ) \frac{(4 − E^{2})^{1/2} dE}{π}  (1.15)

and let {E_{j}^{±}}_{j=1}^{N_{±}} be the point masses of dρ (eigenvalues of J) in (2, ∞) ∪ (−∞, −2). Then (1.13) holds if and only if σ_{ess}(dρ) = [−2, 2], Q(dρ) < ∞, and ∑_{j=1}^{N_{±}}(|E_{j}^{±}| − 2)^{3/2} < ∞.

(5) Nevai’s Conjecture [80, 59]. For OPRL, if ∑_{n=1}^{∞}|a_{n} − 1| + |b_{n}| < ∞, then Z(dρ) < ∞ (Z given by (1.11)).

The five results listed above capture different aspects of the philosophy that the measure is close to the free case if and only if the coefficients are asymptotic to the free ones. In this paper, we study extensions of all these results to perturbations of a periodic sequence of Jacobi or Verblunsky coefficients, that is,

a_{n+p}^{(0)} = a_{n}^{(0)}  b_{n+p}^{(0)} = b_{n}^{(0)}  n ≥ 1  (1.16)

or

α_{n+p}^{(0)} = α_{n}^{(0)} n ≥ 0 (1.17)

and some fixed p ≥ 1. Note that p = 1 is the perturbation of the free case considered above. For simplicity in the OPUC case, we will normally suppose p is even—indeed, the shape of a CMV matrix repeats itself only after shifting by two rows/columns. As explained in Section 15, the situation when p is odd can be reduced to this using sieving. For OPRL, p is arbitrary.

The philosophy described above becomes more subtle when we move to the periodic setting; rather than having a single ‘free operator’ we have a manifold of them (the isospectral torus). Nevertheless—and this is the main thrust of the paper—spectral measures that are close to those of the isospectral torus correspond to coefficients that approach the isospectral torus. One of the key obstructions here is that a sequence of coefficients may approach the isospectral torus without converging to any particular point therein.

In order to make these heuristics precise, we need to make a few definitions. To keep the presentation as coherent as possible, we will focus our attention on the OPRL/Jacobi case for the remainder of the introduction.

To any pair of p-periodic sequences, {a_{n}^{(0)}, b_{n}^{(0)}}_{n∈ℤ}, we can associate a twosided Jacobi matrix J_{0}. Two such pairs of sequences are termed isospectral if the corresponding Jacobi matrices have the same spectrum. We write T_{J_{0}} for the set of p-periodic sequences that are isospectral to J_{0}. Topologically, this is a torus as explained in Subsection 2.14 below.

Given two bounded sequences {a_{n}, b_{n}}_{n=1}^{∞} and {a′_{n}, b′_{n}}_{n=1}^{∞}, we define

d_{m}((a, b), (a′, b′)) = ∑_{k=0}^{∞} e^{−k}[|a_{m+k} − a′_{m+k}| + |b_{m+k} − b′_{m+k}|]  (1.18)

which is a metric for the product topology on ×_{m}^{∞}((0, R] × [−R, R]). The OPUC analog is

d_{m}((α), (α′)) = ∑_{k=0}^{∞} e^{−k}|α_{m+k} − α′_{m+k}|  (1.19)

a metric for ×_{m}^{∞} 𝔻. The distance from a point to a set is defined in the usual way:

d_{m}((a, b), T ) = inf{d_{m}((a, b), (a′, b′)) : (a′, b′) ∈ T }  (1.20)

and similarly in the OPUC case.

We begin with the periodic analog of Weyl’s Theorem.

Theorem 1.1 (Last–Simon [67]). Let J_{0} be a two-sided periodic Jacobi matrix and J a one-sided Jacobi matrix with Jacobi parameters {a_{m}, b_{m}}_{m=1}^{∞}. If

d_{m}((a, b), T_{J_{0}}) → 0  (1.21)

then

σ_{ess}(J) = σ(J_{0})  (1.22)

As indicated, this result first appeared in [67]. It is derived from a theorem that had earlier been proven with different methods by others [41, 69, 85]. The inclusion σ_{ess}(J) ⊃ σ(J_{0}) follows easily using trial vectors; the reverse seems to be more sophisticated. In Section 8, we prove this using the methods of this paper. The OPUC version appears here as Theorem 15.1; it was also proved in [67].

Note that (1.21) does not imply that there is a sequence {(a′_{n}, b′_{n})} ∈ T_{J_{0}} such that

d_{m}((a, b), (a′, b′)) → 0

It is much weaker. Equality of essential spectra under this stronger hypothesis follows immediately from Weyl’s original theorem on compact perturbations.

Our first major new result is an analog of the Denisov–Rakhmanov Theorem.

Theorem 1.2. Let J_{0} be a two-sided periodic Jacobi matrix and J a one-sided Jacobi matrix with Jacobi parameters {a_{m}, b_{m}}_{m=1}^{∞}. If σ_{ess}(J) = σ(J_{0}) and

Σ_{ac}(J) = σ(J_{0})  (1.23)

then d_{m}((a, b), T_{J_{0}}) → 0.

Remark. Using Theorem 1.4 below, we will show that the hypotheses of this theorem can hold while (a, b) only approaches T_{J_{0}} without actually having a limit.

A two-sided p-periodic Jacobi matrix is said to have all gaps open if the spectrum has exactly p connected components—the largest number possible. As explained in Section 2, this is holds generically (indeed, on a dense open set).

Our next new result is

Theorem 1.3. Let J_{0} be a two-sided periodic Jacobi matrix with all gaps open and parameters {a_{n}^{(0)}, b_{n}^{(0)}}_{n=−∞}^{∞}. Also, let J be a one-sided Jacobi matrix with parameters {a_{n}, b_{n}}_{n=1}^{∞} and spectral measure dρ. We assume that σ_{ess}(J) = σ(J_{0}) and

∑_{j=1}^{N} dist(E_{j}, σ_{ess}(J))^{1/2} < ∞  (1.24)

where {E_{m}}_{j=1}^{N} enumerates the eigenvalues of J outside σ(J_{0}). Then

− ∫_{σ(J_{0})} log(\frac{dρ_{ac}}{dx} )dist(x, ℝ \ σ(J_{0}))^{−1/2} dx < ∞  (1.25)

implies

lim_{N→∞} ∑_{j=1}^{pN} log(\frac{a_{j}}{a_{j}^{(0)}} )  (1.26)

exists and lies in (−∞, ∞). Conversely, (1.25) holds so long as

lim sup_{N→∞} ∑_{j=1}^{N} log(\frac{a_{j}}{a_{j}^{(0)}} ) > −∞  (1.27)

and in this case, the limit in (1.26) exists and lies in (−∞, ∞).

Lastly, if (1.25) or (1.27) holds, then

∑_{m=0}^{∞} d_{m}((a, b), T_{J_{0}})^{2} < ∞  (1.28)

and there is J_{1} ∈ T_{J_{0}}, so

d_{m}(J, J_{1}) → 0  (1.29)

Remarks. 1. Thus, when (1.24)–(1.27) hold, J has a limit J_{1} in T_{J_{0}}. In the normal direction to T_{J_{0}}, the convergence is ℓ^{2} (in the sense of (1.28)). But in the tangential direction, we only prove it has a limit. It would be interesting to know what can be said about how slowly (1.29) can occur and to know if there are examples where (1.24)–(1.28) hold but

∑_{m=0}^{∞} d_{m}(J, J_{1})^{2} = ∞  (1.30)

2. Notice that (1.27) will only fail if the partial sums converge to −∞.

3. The final statement that there is J_{1} ∈ T_{J_{0}} with (1.29) is not our result but one of Peherstorfer–Yuditskii [84]. With our methods, we can sometimes prove that the a_{m}’s approach a periodic limit which are the a’s of a point in T_{J_{0}} but we have not been able to control the b’s.

4. By (2.23), all a_{j}^{(0)} in (1.26) and (1.27) can be replaced by Cap(σ(J_{0})), the logarithmic capacity of the spectrum of J_{0}.

Our third new result is

Theorem 1.4. Let J_{0} be a two-sided Jacobi matrix with all gaps open and parameters {a_{n}^{(0)}, b_{n}^{(0)}}_{n∈ℤ}. Let J be a Jacobi matrix with parameters {a_{n}, b_{n}}_{n=1}^{∞}. Then

∑_{m=0}^{∞} d_{m}((a, b), T_{J_{0}})^{2} < ∞  (1.31)

if and only if

(i) σ_{ess}(J) = σ(J_{0}),

(ii) ∑_{j=1}^{N} dist(E_{j}, σ_{ess}(J))^{3/2} < ∞, and

(iii) − ∫_{σ(J_{0})} log(\frac{dρ_{ac}}{dx} )dist(x, ℝ \ σ(J_{0}))^{1/2} dx < ∞.

Here {E_{m}}_{m=1}^{N} enumerates the (discrete) spectrum of J outside σ(J_{0}).

Remarks. 1. Since (i)–(iii) are equivalent to (1.31), one may easily construct examples were (i)–(iii) hold, but there is no J_{1} with (1.29). This provides the examples promised in the remark after Theorem 1.2. It also shows a stark difference between (1.24)–(1.25) and (ii)–(iii). In terms of the spectral measure, this difference is reflected only in the behavior near the band edges.

2. As we will see (Section 12), there are results even if all gaps are not open, but for Theorem 1.4 they are not so easy to express directly in terms of the a’s and b’s.

3. A special case of part of Theorem 1.4 is known, namely, Killip [57] proved that Σ_{ac}(J) = σ(J_{0}) for {a_{n}, b_{n}}_{n=0}^{∞} obeying ∑_{n=1}^{∞}|a_{n} − a_{n}^{(0)}|^{2} + |b_{n} − b_{n}^{(0)}|^{2} < ∞ (which is a strictly stronger hypothesis than (1.31)).

Theorem 1.5. Let J_{0} be a two-sided Jacobi matrix and J a Jacobi matrix with Jacobi parameters {a_{n}, b_{n}}_{n=1}^{∞}. Suppose

∑_{n=1}^{∞} |a_{n} − a_{n}^{(0)}| + |b_{n} − b_{n}^{(0)}| < ∞  (1.32)

Then (1.25) holds.

Remarks. 1. Condition (1.32) can be replaced by

∑_{n=1}^{∞} d_{n}((a, b), T_{J_{0}}) < ∞  (1.33)

Indeed, if (1.33) holds, then (1.32) holds with {a_{n}^{(0)}, b_{n}^{(0)}} replaced by some fixed sequence in T_{J_{0}}.

2. As we will show (see Proposition 3.5), the theorems above continue to hold if d_{m} is replaced by

d̃_{m}((a, b), (a′, b′)) = ∑_{k=0}^{p−1}(|a_{m+k} − a′_{m+k}| + |b_{m+k} − b′_{m+k}|)  (1.34)

For OPUC, we need to sum k from 0 to p in order to get an equivalence; see the discussion at the end of Section 4.

In Section 15, we prove an OPUC analog of each of these theorems. We need to replace the all-gaps-open hypothesis with a stronger one (that holds generically). The deficiency is not so much with our method, but rather that an independent question, which is known in the Jacobi case (independence of the Toda Hamiltonians), is currently unresolved in the CMV case. Our results confirm Conjectures 12.2.3 and 12.2.4 of [99] as well as Conjectures 12.2.5 and 12.2.6 in the (generic) special case that all gaps are open.

For the case of OPUC with a single gap, the analog of Theorem 1.2 is known and motivated Simon’s conjectures in [99]. In that case, the isospectral tori are labelled by a ∈ (0, 1) and consist of {α^{(λ)} : λ ∈ ∂𝔻} where α_{n}^{(λ)} = λa. Then d_{m}(α, T ) → 0 is equivalent to |α_{n}| → a and α_{n+1}/α_{n} → 1. This is often called the López condition. Bello–López [10] proved the OPUC analog of Theorem 1.2 for this case if σ_{ess}(J) = σ(J_{0}) is strengthened to σ(J) = σ(J_{0}) (the analog of Rakhmanov’s result). The full analog for this special case appears in Simon [99], Alfaro et al. [2], and Barrios et al. [9].

Associated to each two-sided p-periodic Jacobi matrix, J_{0}, is a polynomial, Δ_{J_{0}}, of degree p, known as the discriminant. This is a classical object described in detail in the next section. It is usually defined as the trace of the one-period transfer matrix. It is also the unique polynomial (with positive leading coefficient) such that

σ(J_{0}) = {x : Δ_{J_{0}}(x) ∈ [−2, 2]}

In particular, two sequences of coefficients are isospectral if and only if they give rise to the same discriminant.

The key to the proofs of our results is what we call the magic formula. Let J be a two-sided Jacobi matrix, then

Δ_{J_{0}}(J) = S^{p} + S^{−p}  (1.35)

if and only if J ∈ T_{J_{0}}. Here S is the right shift (cf. (2.31)). In particular, (1.35) already implies that J is periodic! In the OPUC case, Δ is a polynomial in z and z^{−1}. It turns out that Δ(C) is always self-adjoint; moreover,

Δ_{C_{0}}(C) = S^{p} + S^{−p}  (1.36)

if and only if C ∈ T_{C_{0}}.

It has been previously noted that for periodic J_{0}, one has

Δ_{J_{0}}(J_{0}) = S^{p} + S^{−p}

That this holds for some polynomial in J_{0} is in Naĭman [76, 77]. That the polynomial is the discriminant was found by Sebbar–Falliero [93]. After learning of our results, L. Golinskii has kindly pointed out to us that Naĭman [77] also has a theorem which implies any J obeying (1.35) is periodic, the core of proving the converse. We will discuss this further in Section 3.

Nonetheless, the two facts that make (1.35) magical to us—that it characterizes the isospectral torus and that it is ideal for the study of perturbations—seem to have escaped prior notice.

While J may be tridiagonal and C five-diagonal, both Δ(J) and Δ(C) are 2p+1diagonal, that is, vanishing except for the main diagonal and p diagonals above and below. Thus, both will be tridiagonal if written as p × p blocks. The key to our proofs will be to extend results from the a_{n} ≡ 1, b_{n} ≡ 0 case to block tridiagonal matrices, and then use (1.35) or (1.36) to study perturbations of the periodic case.

The magic formula is very powerful and opens up many other avenues for study:

(a) Szegő and Jost asymptotics for periodic perturbations and, in particular, the analogs of Damanik–Simon [21].

(b) Periodic analogs of the results of Nevai–Totik [81] and its various recent extensions [22, 97, 101].

(c) Analogs of the Strong Szegő Theorem for the periodic case following Ryckman’s

paper [91] for the Jacobi case. We should point out a major limitation of our results. If B is a disjoint finite

union of closed intervals in ℝ (or ∂𝔻), one can construct an isospectral torus of Jacobi (or CMV) matrices whose recursion coefficients are almost periodic. As discussed in Section 2, these are strictly periodic if and only if the harmonic measure of each interval is rational. There are obvious potential extensions of Theorems 1.1– 1.5 to this setting, but except for Theorem 1.1 (where the method of [67] applies) and Theorem 1.2 (where Section 9 has some extensions), we do not know how to prove them (or even if they are true). There is no analog of Δ in the almost periodic case, so our method does not work directly.

Here is the plan of this paper. Section 2 reviews the theory of the (unperturbed) periodic problem. In Section 3, we prove the magic formula for OPRL, and in Section 4, the magic formula for OPUC. While we will not discuss Schrödinger operators in detail here, we discuss the magic formula for such operators in Section 5. As we have mentioned, the magic formula brings block Jacobi matrices into play, so Section 6 discusses matrix-valued OPRL and OPUC—mainly setting up notation. Section 7 uses known results on Rakhmanov’s theorem for matrix-valued orthogonal polynomials to prove a Denisov-type extension which we use in Section 8 to prove Theorem 1.2; the section also proves half of Theorem 1.1. Section 9 provides two results that go beyond the periodic case to prove Denisov–Rakhmanov-type theorems for special almost periodic situations. Section 10, following [96], proves the P_{2} sum rule of Killip–Simon [59] and the C_{1} sum rule for matrix-valued measures, and Section 11 uses these results to prove Theorems 1.3 and 1.4. Section 12 explores what we can say if gaps are closed. Section 13 proves analogs to the Lieb–Thirring bounds of Hundertmark–Simon [53] as preparation for proving Theorem 1.5 in Section 14. Finally, Section 15 discusses the OPUC results.

Acknowledgements. It is a pleasure to thank Leonid Golinskii, Irina Nenciu, Leonid Pastur, and Peter Yuditskii for useful discussions.

2. Review of the Periodic Problem

In this section, we collect some of the major elements in the strictly periodic case. As this is textbook material, we forgo proofs and historical discussion. Full details can be found, for example, in Chapter 11 of [99] or Chapter 5 of [102] and the references therein.

To discuss the strictly periodic case, we need to extend our operators to be twosided, that is, to act on ℓ^{2}(ℤ). In the Jacobi/OPRL case, we simply continue the tridiagonal pattern with parameters {a_{n}, b_{n}}_{n∈ℤ}. Two-sided (or extended) CMV matrices are formed as C = LM, where L and M are doubly infinite direct sums

L = ⋯ ⊕ Θ_{−2}(α_{−2}) ⊕ Θ_{0}(α_{0}) ⊕ Θ_{2}(α_{2}) ⊕ ⋯  (2.1)
M =  ⋯ ⊕ Θ_{−1}(α_{−1}) ⊕ Θ_{1}(α_{1}) ⊕ ⋯  (2.2)

that are misaligned by one row/column, just as in (1.5)–(1.8).

We adopt the convention of indexing the elements of matrices so that

J_{00} = b_{1}  L_{00} = ᾱ_{0}  M_{00} = −α_{−1}  (2.3)

except M_{00} = 1 in the one-sided case.

2.1. Transfer Matrices. Let J be a two-sided Jacobi matrix. A sequence {u_{n}} obeys (J − x)u ≡ 0 if and only if

a_{n+1}u_{n+1} + (b_{n+1} − x)u_{n} + a_{n}u_{n−1} = 0  (2.4)

or, what is equivalent,

(\begin{matrix}u_{n+1} \\ a_{n+1}u_{n} \end{matrix}) = A_{n} (\begin{matrix}u_{n} \\ a_{n}u_{n−1} \end{matrix})  (2.5)

with

A_{n}(x) = \frac{1}{a_{n+1}} (\begin{matrix}x − b_{n+1} & −1 \\ a_{n+1}^{2} & 0 \end{matrix})  (2.6)

Note that the desire to have A_{n} depend only on one pair (a_{n+1}, b_{n+1}) and have determinant equal to one resulted in the factors a_{n+1} and a_{n} appearing in (2.5). (The same price is usually paid when writing Sturm–Liouville equations as first-order systems.) The choice is not the most common one (although it is used in Pastur– Figotin [83]), but we feel it is the ‘right’ one since, in particular, det(A_{n}(x)) = 1.

In the OPUC case we define

A_{n}(z) = \frac{1}{ρ_{n}} (\begin{matrix}z & −ᾱ_{n} \\ −α_{n}z & 1 \end{matrix})  (2.7)

which encodes the recurrence relation (1.2):

(\begin{matrix}φ_{n+1}(z) \\ φ_{n+1}^{∗}(z) \end{matrix}) = A_{n} (\begin{matrix}φ_{n}(z) \\ φ_{n}^{∗}(z) \end{matrix})  (2.8)

We will now explain the link to formal (i.e., not necessarily ℓ^{2}) eigenvectors of a two-sided CMV matrix, C. It is not as simple as (2.5). Suppose (C − z)u = 0 with z ≠ 0 and let v = Z^{−1}Mu where Z denotes the diagonal matrix with entries

Z_{jj} = \begin{cases}z & : j odd \\ 1 & : j even \end{cases}

and M is as in (2.2). Then

z (\begin{matrix}u_{2n+2} \\ v_{2n+2} \end{matrix}) = zA_{2n+1}(z) (\begin{matrix}v_{2n+1} \\ u_{2n+1} \end{matrix}) = A_{2k+1}(z)A_{2k}(z) (\begin{matrix}u_{2n} \\ v_{2n} \end{matrix})  (2.9)

The key observation used to verify (2.9) is

(\begin{matrix}x′ \\ y′ \end{matrix}) = A_{n}(z) (\begin{matrix}y \\ x \end{matrix}) ⟺ (\begin{matrix}zy \\ y′ \end{matrix}) = Θ(α_{n}) (\begin{matrix}x \\ x′ \end{matrix})  (2.10)

2.2. The Discriminant. As in Sturm–Liouville theory, the discriminant is defined as the trace of the one-period transfer matrix:

Δ(z) = Tr(T (z))  (2.11)

where

T (z) = \{ \begin{matrix}A_{p−1}(z) ⋯ A_{1}(z)A_{0}(z) & (OPRL) \\ z^{−p/2}A_{p−1}(z) ⋯ A_{1}(z)A_{0}(z) & (OPUC) \end{matrix}  (2.12)

In the OPUC case, p is even. Also, the factor z^{−p/2} is there to cancel the extra factor of z on the left-hand side of (2.9). From a strictly OPUC point of view, it is more natural to omit this factor (as in [98, 99]); however, as the magic formula is an operator identity, we have elected to use the definition best adapted to this perspective. The only negative side effect of this choice is that our Lyapunov exponent (defined below) differs by −\frac{1}{2} log |z| from that in [98, 99].

For OPRL, the discriminant is a real polynomial of degree p with leading behavior

Δ(x) = (a_{1} … a_{p})^{−1}[ ∏_{j=1}^{p}(x − b_{j}^{(0)}) + O(x^{p−2})]  (2.13)

= (a_{1} ⋯ a_{p})^{−1}[x^{p} − ( ∑_{j=1}^{p} b_{j}^{(0)})x^{p−1} + O(x^{p−2})]  (2.14)

For OPUC, it is a Laurent polynomial of total degree p with

Δ(z̄) = \overline{Δ(1/z)}  (2.15)

so Δ is real-valued on ∂𝔻. Moreover,

Δ(z) = (ρ_{0}ρ_{1} ⋯ ρ_{p−1})^{−1}(z^{p/2} + ⋯ + z^{−p/2})  (2.16)

2.3. The Lyapunov Exponent. On an exponential scale, the behavior of formal eigenfunctions is determined by the Lyapunov exponent

γ(z) = lim_{n→∞} \frac{1}{np} log ∥T^{n}(z)∥  (2.17)
= \frac{1}{p} log (spectral radius of T (z))
= \frac{1}{p} log |λ_{+}(z)|  (2.18)

where λ_{±} are the eigenvalues of T (z) with the convention |λ_{+}| ≥ |λ_{−}|. As det(T (z)) ≡ 1, these eigenvalues are the roots of

λ^{2} − Δ(z)λ + 1 = 0  (2.19)

which implies

λ_{±}(z) = \frac{Δ(z)}{2} ± \frac{\sqrt{Δ^{2}(z) − 4}}{2}  (2.20)

and so

γ(z) = \frac{1}{p} log |\frac{1}{2}Δ(z) + \frac{1}{2} \sqrt{Δ(z)^{2} − 4}|  (2.21)

2.4. Gaps and Bands. Our recurrence relations admit bounded solutions for a given z if and only if Δ(z) ∈ [−2, 2]. In the Jacobi/OPRL case, this is a collection of intervals in ℝ. For CMV/OPUC, it is a collection of arcs in ∂𝔻. In either case, one may partition this set into p bands. These are the closures of the (disjoint) regions where Δ(z) ∈ (−2, 2). These can only intersect at the ‘band edges’, Δ^{−1}({−2, 2}).

The open gaps are the intervals/arcs that are complementary to the bands— excluding the two semi-infinite intervals in the OPRL case. When two bands touch, we refer to the common band edge as a closed gap.

Δ^{2} − 4 has simple zeros at the edges of the open gaps and double zeros at the closed gaps; indeed, this is a complete list of its zeros. It is possible to distinguish whether these zeros correspond to Δ(z) = ±2 from the fact that there must be two zeros of Δ ± 2 between consecutive zeros of Δ ∓ 2 and the fact that Δ has positive leading coefficient.

2.5. Spectrum. In both cases, the spectrum of the two-sided operator (acting on ℓ^{2}(ℤ)) is the union of the bands: σ = Δ^{−1}([−2, 2]). It is purely absolutely continuous and of multiplicity two.

The spectrum of a two-sided p-periodic operator uniquely determines its discriminant; see Lemma 3.3. One consequence of this was noted already in the introduction: isospectral tori are the classes of p-periodic recurrence coefficients that lead to the same discriminant.

In the case of a one-sided operator, the essential spectrum remains Δ^{−1}([−2, 2]); it is absolutely continuous with multiplicity one. In addition, up to one eigenvalue may appear in each open gap.

2.6. Potential Theory. From the way they are defined, one can see that γ(z) vanishes on the bands and is both positive and harmonic in the complement (in the OPUC case one must also exclude z = 0). This leads to the solution of the Dirichlet problem for a charge at infinity,

g_{ℂ\σ}(z; ∞) = \{ \begin{matrix}\frac{1}{p} log |\frac{Δ}{2} + \sqrt{\frac{Δ^{2}}{4} − 1}| & (OPRL) \\ \frac{1}{2} log|z| + \frac{1}{p} log |\frac{Δ}{2} + \sqrt{\frac{Δ^{2}}{4} − 1}| & (OPUC) \end{matrix}  (2.22)

and so to the logarithmic capacity of the spectrum,

Cap(σ) = \begin{cases}(∏_{j=1}^{p} a_{j})^{1/p} & (OPRL) \\ (∏_{j=0}^{p−1} ρ_{j})^{1/p} & (OPUC) \end{cases}  (2.23)

2.7. Harmonic Measure. Taking normal derivatives in (2.22) leads to a formula for harmonic measure on σ (aka equilibrium measure for the logarithmic potential),

dν = \begin{cases}\frac{2}{p} \frac{|Δ′(x)|}{\sqrt{4−Δ^{2}(x)}} \frac{dx}{2π} & (OPRL) \\ \frac{2}{p} \frac{|Δ′(e^{iθ})|}{\sqrt{4−Δ^{2}(e^{iθ})}} \frac{dθ}{2π} & (OPUC) \end{cases}  (2.24)

where supp(dν) = σ = {z : |Δ(z)| ≤ 2}. (Note: [99] has 1/p rather than 2/p, but that is an error.)

Recognizing

\frac{Δ′(x)}{\sqrt{4 − Δ^{2}(x)}} = \frac{d}{dx} arccos(\frac{Δ(x)}{2})  (2.25)

we see that the harmonic measure of each band is exactly 1/p. The connected components of the union of the components all have rational harmonic measure. This gives strong restrictions on sets that can be bands. In the OPRL case, rational harmonic measure of connected components is also sufficient for a set to be the spectrum of a periodic Jacobi matrix. In the OPUC case, there is an additional condition needed: after breaking the bands into arcs of harmonic measure 1/p, the harmonic midpoints {ζ_{j}}_{j=1}^{p} of these intervals must obey ∏_{j=1}^{p} ζ_{j} = 1. Clearly, the condition on the harmonic midpoints can be achieved by simply rotating σ. Discarding this condition gives rise to Verblunsky coefficients that are p-automorphic, α_{n+p} = e^{iϕ}α_{n}, rather than p-periodic.

2.8. Thouless Formula. Harmonic measure appears naturally in the theory in several other ways. It is the density of states measure:

lim_{N→∞} \frac{1}{2N} ∑_{n=−N}^{N} f(J)_{nn} = ∫ f(x) dν(x)  (2.26)

for every polynomial (or continuous function) f. The same formula holds with C replacing J. This connection, or more precisely the resulting expression for γ(z) in terms of Cap(σ) and the logarithmic potential of dν, is known as the Thouless formula.

Two further characterizations of dν involve the orthogonal polynomials. dν is the weak limit of \frac{1}{N} ∑_{n=0}^{N−1} p_{n}^{2}(x) dρ(x) (resp. \frac{1}{N} ∑_{n=0}^{N−1}|φ_{n}(e^{iθ})|^{2}dμ(θ)). It is also the limiting density of zeros, that is, the weak limit of the probability measures, dν_{n}, which give weight 1/n to each of the zeros of p_{n} (resp. φ_{n}). These two characterizations are closely linked to (2.26); however, in the OPUC case one should keep in mind that for each n, the zeros of ϕ_{n} lie strictly inside 𝔻.

2.9. Floquet Theory. Looking at the eigenvalues of T, one sees that when λ_{+} ≠ λ_{−}, there is a basis of formal (i.e., non-ℓ^{2}) eigenfunctions obeying

u_{m+kp} = λ_{±}^{k}u_{m}  (2.27)

If λ_{+} = λ_{−}, which happens precisely at the band edges, then both are ±1. If the edge abuts an open gap, there is only one eigenfunction obeying (2.27) since T has a Jordan block structure. At closed gaps, T = ±1 and so all solutions are p-periodic.

Solutions obeying

u_{m+kp} = e^{ikθ}u_{m}  (2.28)

are called Floquet solutions and e^{iθ} is called the Floquet index; they have much the same role as plane waves in Fourier analysis. Since λ_{−} = λ_{+}^{−1}, if (2.28) has a solution, then e^{−iθ} is also a Floquet index.

In the OPRL case,

(2.28) holds ⟺ Δ(x) = 2 cos θ  (2.29)

Thus, by the discussion above, for each θ ∈ (0, π), (2.28) or (2.29) holds for exactly p values of x: x_{1}(θ) < x_{2}(θ) < ⋯ < x_{p}(θ). These x_{j}(θ) are known as the band functions.

The changes in the OPUC case are purely notational:

(2.28) holds ⟺ Δ(z) = 2 cos θ  (2.30)

For θ ∈ (0, π), this has p solutions all of which lie in ∂𝔻.

2.10. Direct Integrals. Let S denote the right shift,

(Su)_{n} = u_{n−1}  (2.31)

If the sequences of coefficients are p-periodic, then J (or C) commutes with S^{p}, which means that the two operators can be ‘simultaneously diagonalized’. We elaborate this point in the OPRL case; the OPUC is almost identical.

Let us write

H_{p} := ∫ ^{⊕} ℓ_{θ}^{2} \frac{dθ}{2π} = L^{2}([0, 2π), \frac{dθ}{2π}; ℂ^{p})

where ℓ_{θ}^{2} is the p-dimensional Hilbert space

ℓ_{θ}^{2} = {u|u_{n+p} = e^{iθ}u_{n}} ⟨u|v⟩_{θ} = ∑_{n=1}^{p} ū_{n}v_{n}

From Fourier analysis, there is a unitary operator F: ℓ^{2}(ℤ) → H_{p} so that FS^{p}F^{−1} is multiplication by e^{iθ}1 and FJF^{−1} acts fiber-wise (i.e., on each ℓ_{θ}^{2}) as a p × p matrix, J(θ). In particular, the eigenvalues of J(θ) are the solutions of (2.29), that is, they are the band functions x_{j}(θ).

2.11. Hyperelliptic Riemann Surfaces. As \sqrt{Δ^{2} − 4} appears repeatedly in the theory, it is natural that the associated Riemann surface should enter the analysis. Δ^{2} − 4 has simple zeros at the edges of open gaps and at inf σ(J) and sup σ(J). It has double zeros at the closed gaps. Let ℓ denote the number of open gaps, then \sqrt{Δ^{2} − 4} has square root singularities at 2(ℓ + 1) points, and so its natural analyticity domain is the genus ℓ Riemann surface, S, obtained by taking two copies of ℂ \ σ(J), gluing at the bands and adding points at ∞. There is a natural projection π: S → ℂ∪\{∞\} which is 2 to 1 except at the branch points of \sqrt{Δ^{2} − 4}. A similar analysis works for OPUC, but now there are ℓ gaps and the genus is ℓ−1.

2.12. Minimal Herglotz and Carathéodory Functions. For a half-line periodic Jacobi matrix, the m function is defined by

m(z) = ⟨δ_{0}, (J − z)^{−1}δ_{0}⟩  ∀z ∈ ℂ \ σ(J)  (2.32)

This can be shown to obey a quadratic equation with polynomial coefficients

A(z)m(z)^{2} + B(z)m(z) + C(z) = 0  (2.33)

Moreover, these coefficients can be chosen to obey

B^{2} − 4AC = Δ^{2} − 4  (2.34)

This implies that m(z) has meromorphic continuation to S. Indeed, m has minimal degree (i.e., degree ℓ + 1 in the OPRL case and ℓ in the OPUC case) among all meromorphic functions on S that are not of the form g ∘ π with g meromorphic on the Riemann sphere. It can be shown that there is a one-one correspondence between minimal meromorphic functions obeying Im m(z) > 0 if Im z > 0 and m(z) = −z^{−1} + O(z^{−2}) on the top sheet of S and all periodic Jacobi parameters with the same Δ. (We call these minimal Herglotz functions.)

There is a similar description for OPUC, but now one uses

F (z) = ⟨δ_{0}, (C + z)(C − z)^{−1}δ_{0}⟩  (2.35)

which obeys Re F (z) > 0 if |z| < 1 and F (0) = 1. Again F obeys a quadratic equation, showing that F has a meromorphic continuation to S of minimal degree, and again there is a one-one correspondence between all {α_{n}}_{n=0}^{p−1} with the same Δ and all minimal Carathéodory functions.

2.13. Dirichlet Data. One can describe the set of minimal Herglotz functions in terms of their poles. For each open gap, \{G_{j}\}_{j=1}^{ℓ}, π^{−1}(\overline{G}_{j}) ≡ T_{j} is a circle since π is 2 to 1 on G_{j} and one-one on \overline{G}_{j} \ G_{j}. A meromorphic Herglotz function has ℓ + 1 simple poles, one at ∞ on the second sheet and the other ℓ, one in each T_{j}. Thus, the set of meromorphic Herglotz functions is homeomorphic to ×_{j=1}^{ℓ} T_{j} under the bijective map from such functions to its poles. A similar analysis holds for OPUC but now there is no pole at infinity, there are ℓ gaps, and ×_{j=1}^{ℓ} T_{j} describes the possible poles. The difference is that for OPRL, the dimension of the torus is ℓ, and for OPUC it is ℓ − 1.

2.14. Isospectral Tori. By combining the bijective maps from periodic OPRL to minimal Herglotz functions and of such functions to Dirichlet data, we see for a Δ of period p with ℓ gaps,

{(a_{n}, b_{n})_{n=1}^{p} : the discriminant is Δ}

is an ℓ-dimensional torus in ℝ^{2p}. Generically, ℓ = p − 1. In the OPUC case, generically ℓ = p and the torus is naturally embedded in ℂ^{p}. This torus is the isospectral torus which we will denote by T or T_{J_{0}} if a given periodic J_{0} underlies our construction. For clarity of exposition, we will typically blur the distinction between p-tuples (a_{n}, b_{n})_{n=1}^{p} and the corresponding infinite sequences {a_{n}, b_{n}}_{n∈ℤ} of period p.

2.15. Isospectral Flows. The fact that spaces of p-periodic coefficients foliate into tori suggests that there is some kind of completely integrable system in the background. That is true: it is the Toda flow in the OPRL case and the defocusing Ablowitz–Ladik flow in the OPUC case. Since we will not need these below, we say no more about them, but see Chapter 6 of [102] for the OPRL case and Section 11.11 of [99] for OPUC.

3. The Magic Formula for Jacobi Matrices

Our goal in this section is to prove

Theorem 3.1. Let J_{0} be a two-sided p-periodic Jacobi matrix with discriminant Δ_{J_{0}}(x) and isospectral torus T_{J_{0}}. Let J be a two-sided (not a priori periodic) Jacobi matrix. Then

Δ_{J_{0}}(J) = S^{p} + S^{−p} ⟺ J ∈ T_{J_{0}}  (3.1)

where S is the right shift, (2.31), on ℓ^{2}(ℤ).

We provide two proofs of the ‘harder’ direction ⇒ or rather of

Δ_{J_{0}}(J) = S^{p} + S^{−p} ⇒ J is periodic  (3.2)

which is the key step. Our first proof is immediately below; the second, suggested to us by L. Golinskii, appears after Lemma 3.4.

Lemma 3.2. Let ℓ = 1, 2, . . .. Then

(J^{ℓ})_{m,m+k} = \begin{cases}0 & k > ℓ \\ a_{m}a_{m+1} ⋯ a_{m+k−1} & k = ℓ \\ a_{m}a_{m+1} ⋯ a_{m+ℓ−2}(b_{m} + b_{m+1} + ⋯ + b_{m+ℓ−1}) & k = ℓ − 1 \end{cases} (3.3)

Proof. Writing

(J^{ℓ})_{m,m+k} = ∑_{i_{1},...,i_{ℓ−1}} J_{m,i_{1}}J_{i_{1},i_{2}} ⋯ J_{i_{ℓ−1},m+k}  (3.4)

We see that since J is tridiagonal, all terms are zero if k > ℓ, that we must have (with i_{0} ≡ m, i_{ℓ} = m + k) that i_{q} − i_{q−1} = 1 for q = 1, …, ℓ if k = ℓ, and that if k = ℓ − 1, i_{q} − i_{q−1} = 1 for all but one q ∈ {1, …, ℓ} and it is zero for that q. □

Lemma 3.3. If J and J_{0} are periodic, then σ(J) = σ(J_{0}) if and only if Δ_{J} = Δ_{J_{0}}.

Remark. This lemma says that the spectrum determines the discriminant and vice versa. That the discriminant determines the spectrum is elementary: σ = {x : Δ(x) ∈ [−2, 2]}. Therefore we only prove the other direction—indeed, we give two proofs.

First Proof. Harmonic measure dν is intrinsic to the set σ; it is the solution of an electrostatic problem there. But then dν determines Δ via (2.24). □

Second Proof. σ determines the gaps—even closed gaps—via harmonic measure. The gap edges determine the zeros of Δ − 2 and so Δ − 2 up to a constant. The zeros of Δ + 2 then determine the constant. □

Proof of Theorem 3.1. For all θ ∈ [0, 2π), J(θ) is self-adjoint and so diagonalizable. Moreover, the eigenvalues of J_{0}(θ) are precisely the roots of Δ(x) = 2 cos(θ). Thus

Δ_{J_{0}}(J_{0}(θ)) = (2 cos θ)1  (3.5)

But then Δ_{J_{0}}(J_{0}) and S^{p} +S^{−p} both have direct integral decomposition with fibers (2 cos θ)1, so

Δ_{J_{0}}(J_{0}) = S^{p} + S^{−p}  (3.6)

Since J ∈ T_{J_{0}} ⇒ Δ_{J} = Δ_{J_{0}}, this proves ⇐ in (3.1).

Now suppose LHS of (3.1) holds. By (2.14), (3.3), and

(S^{p} + S^{−p})_{m,m+p} = 1  (S^{p} + S^{−p})_{m,m+p−1} = 0  (3.7)

this implies

a_{m} ⋯ a_{m+p−1} = a_{1}^{(0)} ⋯ a_{p}^{(0)}  (3.8)

and

∑_{j=0}^{p−1} b_{m+j} = ∑_{j=0}^{p−1} b_{j+1}^{(0)}  (3.9)

In particular,

a_{m} ⋯ a_{m+p−1} = a_{m+1} ⋯ a_{m+p}  ∑_{j=0}^{p−1}(b_{m+j+1} − b_{m+j}) = 0

which lead to

a_{m} = a_{m+p}  b_{m} = b_{m+p}  (3.10)

so J is periodic.

Since J is periodic, Δ_{J}(J) = S^{p} + S^{−p}; moreover, Δ_{J_{0}}(J) = S^{p} + S^{−p} by hypothesis. Thus we learn that applying the polynomial Δ_{J} − Δ_{J_{0}} to J gives zero. By the k = ℓ case of (3.3), it must therefore be the zero polynomial, that is, Δ_{J} = Δ_{J_{0}}. Lemma 3.3 now completes the proof. □

Remarks. 1. Showing that J was periodic only required equality in Δ_{J_{0}}(T ) = S^{p} + S^{−p}, for the two most extreme upper (or lower) diagonals. Nevertheless, J ∈ T_{J_{0}} requires equality everywhere.

2. We need not suppose a priori that each a_{n} > 0 and can allow some a_{n} = 0 (J can still be defined on ℓ^{2}(ℤ)), for (3.8) implies that if LHS of (3.1) holds, then each a_{n} > 0.

We now turn to our second proof of (3.2).

Lemma 3.4 (Naĭman [77]). Let A be a two-sided (bounded) infinite matrix of finite width (i.e., for some w, we have that |k − ℓ| > w ⇒ A_{kℓ} = 0). Suppose

[A, S^{p} + S^{−p}] = 0  (3.11)

for some p, then

[A, S^{p}] = 0  (3.12)

Remarks. 1. This is Lemma 2 in [77]; no proof is given.

2. [A, B] ≡ AB − BA

3. (3.12) has an equivalent form:

[A, S^{p}] = 0 ⟺ A_{k+p,ℓ+p} = A_{k,ℓ}  for all k, ℓ  (3.13)

4. As (3.13) shows, [J, S^{p}] = 0 for a Jacobi matrix if and only if a_{k} and b_{k} are p-periodic.

Proof. Since A has finite width, we can find diagonal matrices D_{k_{1}}, D_{k_{1}+1}, …, D_{k_{2}} with D_{k_{1}} ≠ 0 ≠ D_{k_{2}}, so that

A = ∑_{j=k_{1}}^{k_{2}} D_{j}S^{j}  (3.14)

Since D_{j} is diagonal, so is S^{p}D_{j}S^{−p}. Thus

(S^{p} + S^{−p})A = ∑_{j=k_{1}}^{k_{2}}(S^{p}D_{j}S^{−p})S^{j+p} + ∑_{j=k_{1}}^{k_{2}}(S^{−p}D_{j}S^{p})S^{j−p}

A(S^{p} + S^{−p}) = ∑_{j=k_{1}}^{k_{2}} D_{j}S^{j+p} + ∑_{j=k_{1}}^{k_{2}} D_{j}S^{j−p}

Since the composition (3.14) uniquely determines each D_{j}, (3.11) implies

S^{p}D_{k_{2}}S^{−p} = D_{k_{2}}

that is, D_{k_{2}} is periodic. Thus, D_{k_{2}}S^{p} commutes with S^{p} +S^{−p}, so we can remove it from (3.14) without losing (3.11). This shows inductively that each D_{j} is periodic.

□

Second proof of (3.2). J commutes with Δ(J), so (3.12) holds. □

Our next goal is to compare d̃_{m}((a, b), T ) given by (1.34) and d_{m}((a, b), T ) given by (1.18). As well as satisfying natural curiosity, this relation also plays an important role (via Theorem 11.12) in the proofs of Theorems 1.3 and 1.4.

To capture the essence of what follows, let us pause to ponder the following: suppose d̃_{m}((a, b), T ) = 0 for all m, does this mean that (a, b) ∈ T ? The hypothesis tells us that each length-p block belongs to the isospectral torus; it does not a priori guarantee that the coefficients are periodic. Example 4.5 shows that this can occur in the OPUC case; however it does not occur for OPRL. The reason is simple: within the isospectral torus, a_{1}, …, a_{p−1} determines a_{p} and b_{1}, …, b_{p−1} determines b_{p}.

Proposition 3.5. Given a p-periodic Jacobi matrix J_{0}, 1 ≤ q ≤ ∞, and ε > 0, there is a constant C so that

e^{1−p}∥d̃_{m}((a, b), T_{J_{0}})∥_{ℓ^{q}}≤ ∥d_{m}((a, b), T_{J_{0}})∥_{ℓ^{q}}≤ C∥d̃_{m}((a, b), T_{J_{0}})∥_{ℓ^{q}}  (3.15)

for all sequences {(a_{n}, b_{n})} obeying ε^{−1} > a_{n} > ε > 0. All ℓ^{q} norms are over m ∈ {1, 2, 3, …}.

The key input is

Lemma 3.6. Given {(a_{n}, b_{n})} obeying ε^{−1} > a_{n} > ε > 0,

|a_{n} − a_{n}^{(0)}| + |b_{n} − b_{n}^{(0)}| ≤ d̃_{m}((a, b), (a^{(0)}, b^{(0)})) + C ∑_{r=m}^{n−p+1} d̃_{r}((a, b), T_{J_{0}})

for all n ≥ m. The constant C depends only on ϵ.

Proof. The proof is by induction on n. For m ≤ n ≤ m + p, the sum disappears and the result is immediate from the definition of d̃_{m}.

For n > m + p, we consider the functions

f(a_{1}, …, a_{p}) := ∑_{j=1}^{p}[log(a_{j}) − log(a_{j}^{(0)})] g(b_{1}, …, b_{p}) := ∑_{j=1}^{p}[b_{j} − b_{j}^{(0)}]

These vanish on T_{J_{0}}, as explained in the proof of Theorem 3.1.

As g is Lipschitz (with constant 1),

|b_{n} − b_{n−p}| = |g(b_{n}, …, b_{n−p+1}) − g(b_{n−1}, …, b_{n−p})|
≤ |g(b_{n}, …, b_{n−p+1})| + |g(b_{n−1}, …, b_{n−p})|
≤ d̃_{n−p+1}((a, b), T_{J_{0}}) + d̃_{n−p}((a, b), T_{J_{0}})

In a similar way,

|log[a_{n}] − log[a_{n−p}]| = |f(a_{n}, …, a_{n−p+1}) − f(a_{n−1}, …, a_{n−p})|

leads to

|a_{n} − a_{n−p}| ≤ C_{ϵ}[d̃_{n−p+1}((a, b), T_{J_{0}}) + d̃_{n−p}((a, b), T_{J_{0}})]

Combining these two inequalities gives

|a_{n} − a_{n}^{(0)}| + |b_{n} − b_{n}^{(0)}| ≤ |a_{n−p} − a_{n−p}^{(0)}| + |b_{n} − b_{n−p}|

+ (1 + C_{ε})[d̃_{n−p+1}((a, b), T_{J_{0}}) + d̃_{n−p}((a, b), T_{J_{0}})]

which completes the proof of the inductive step. □

Proof of Proposition 3.5. The left-hand inequality in (3.15) follows immediately from the definitions of d_{m} and d̃_{m}; we focus on the second inequality.

Choose (a^{(0)}, b^{(0)}) minimizing d_{m}((a, b), T_{J_{0}}); strictly, this amounts to a (inconsequential) change in J_{0}. Applying Lemma 3.6 in the definition of d_{m} gives

d_{m}((a, b), T_{J_{0}}) ≤ \frac{e}{e−1} d̃_{m}((a, b), T_{J_{0}}) + C ∑_{k=0}^{∞} ∑_{r=m}^{m+k} e^{−k} d̃_{r}((a, b), T_{J_{0}})

≤ C′ ∑_{j=0}^{∞} e^{−j} d̃_{m+j}((a, b), T_{J_{0}})

The proposition follows because convolution with e^{−j}χ_{[0,∞)}(j) is a bounded operator on all ℓ^{q} spaces. □

4. The Magic Formula for CMV Matrices

Our goal in this section is to prove

Theorem 4.1. Let p be even and let C_{0} be a two-sided p-periodic CMV matrix with discriminant Δ_{C_{0}}(z) and isospectral torus T_{C_{0}}. Given a two-sided (not a priori periodic) CMV matrix, C,

Δ_{C_{0}}(C) = S^{p} + S^{−p} ⟺ C ∈ T_{C_{0}}  (4.1)

Remarks. 1. Notice that since C is unitary and Δ(e^{iθ}) is real, Δ_{C_{0}}(C) is self-adjoint.

2. By (2.16) and the fact that C is five-diagonal, Δ_{C_{0}}(C) has 2(p/2) diagonals above/below the main diagonal.

3. As in Section 3, we will first present our initial proof that

Δ_{C_{0}}(C) = S^{p} + S^{−p} ⇒ {α_{n}} is periodic  (4.2)

and then a proof based on Golinskii’s suggestion.

Lemma 4.2. We have:

(C^{ℓ})_{m,m+k} = (C^{−ℓ})_{m,m+k} = 0  if k > 2ℓ  (4.3)
(C^{ℓ})_{2m,2m+2ℓ} = ρ_{2m}ρ_{2m+1} … ρ_{2m+2ℓ−1}  (4.4)
(C^{ℓ})_{2m+1,2m+2ℓ+1} = 0  (4.5)
(C^{−ℓ})_{2m,2m+2ℓ} = 0  (4.6)
(C^{−ℓ})_{2m+1,2m+ℓ+1} = ρ_{2m+1}ρ_{2m+2} … ρ_{2m+2ℓ}  (4.7)
(C^{ℓ})_{2m,2m+2ℓ−1} = ρ_{2m}ρ_{2m+1} … ρ_{2m+2ℓ−2}ᾱ_{2m+2ℓ−1}  (4.8)
(C^{ℓ})_{2m+1,2m+2ℓ} = −α_{2m}ρ_{2m+1} … ρ_{2m+2ℓ−1}  (4.9)
(C^{−ℓ})_{2m,2m+2ℓ−1} = −ᾱ_{2m−1}ρ_{2m}ρ_{2m+1} … ρ_{2m+2ℓ−2}  (4.10)
(C^{−ℓ})_{2m+1,2m+2ℓ} = ρ_{2m+1} … ρ_{2m+2ℓ−1}α_{2m+2ℓ}  (4.11)

Proof. As L and M are tridagonal, C^{ℓ} is a product of 2ℓ tridiagonal matrices, so (4.3) is immediate.

We will prove the results for C^{ℓ}. The results for C^{−ℓ} are similar if we note

Θ(α)^{−1} = Θ(ᾱ)  (4.12)

since Θ is unitary and symmetric.

Equation (4.4) follows from

L_{2m,2m+1} = ρ_{2m}  M_{2m+1,2m+2} = ρ_{2m+1}  (4.13)

and (4.5) from

L_{2m+1,2m+2} = 0  (4.14)

Because of (4.14), the only way for C^{ℓ} to get from 2m to 2m + 2ℓ − 1 is to increase index in the first 2ℓ − 1 factors, which leads to (4.8). For the same reason, to get from 2m + 1 to 2m + 2ℓ, the last 2ℓ − 1 factor must increase index, leading to (4.9). □

Lemma 4.3. If C and C_{0} are p-periodic, then σ(C) = σ(C_{0}) if and only if Δ_{C} = Δ_{C_{0}}.

Proof. Either proof of Lemma 3.3 carries over with no change. □

Proof of Theorem 4.1. The proof that

Δ_{C_{0}}(C_{0}) = S^{p} + S^{−p}  (4.15)

is identical to the proof of (3.5).

For the converse, suppose

Δ_{C_{0}}(C) = S^{p} + S^{−p}  (4.16)

In particular,

Δ_{C_{0}}(C)_{2m,2m+p−1} = 0  (4.17)

By (2.16) and Lemma 4.2, this implies (recall p is even)

(ρ_{0}^{(0)} … ρ_{p−1}^{(0)})^{−1}(ρ_{2m}ρ_{m+1} … ρ_{2m+p−2})(ᾱ_{2m+p−1} − ᾱ_{2m−1}) = 0

so

α_{2m+p−1} = α_{2m−1}  (4.18)

Similarly, since

Δ_{C_{0}}(C)_{2m+1,2m+p} = 0

we get

(ρ_{0}^{(0)} … ρ_{p−1}^{(0)})^{−1}(ρ_{2m+1} … ρ_{2m+p−1})(α_{2m+2} − α_{2m+p−1}) = 0

which leads to

α_{2m+p} = α_{2m}  (4.19)

Thus, α has period p. That C ∈ T_{C_{0}} follows from Lemma 4.3 and the same argument used in the OPRL case. □

Next, we give a proof using Naĭman’s lemma. We will need

Lemma 4.4. Let C be the extended CMV matrix associated to {α_{n}}_{n=−∞}^{∞}. Let p be even. If S^{p}C = CS^{p}, then

α_{n+p} = α_{n}  (4.20)

for all n.

Proof. We have that C_{2j 2j+1}^{2}+C_{2j 2j+2}^{2} = ρ_{2j}^{2} (see (4.2.14) of [98]), so ρ_{2j} is periodic. Thus, C_{2j 2j+2}/ρ_{2j} = ρ_{2j+1} is also periodic. So ᾱ_{2j+1} = C_{2j 2j+1}/ρ_{2j} is periodic as is α_{2j} = C_{2j+1 2j+2}/(−ρ_{2j+1}). □

Second proof that (4.2) holds. C commutes with S^{p} + S^{−p}, so by Naĭman’s lemma (Lemma 3.4), which did not require that A be self-adjoint, S^{p}C = CS^{p}, which implies α is periodic by Lemma 4.4. □

We now turn to the OPUC version of Proposition 3.5. As noted in the introduction, it is not sufficient to sum over exactly one period:

Example 4.5. (0, \frac{1}{2}, 0, \frac{1}{2}, 0, …) and (0, −\frac{1}{2}, 0, −\frac{1}{2}, 0, …) are in the same isospectral torus, namely, the one with p = 2 and

Δ(z) = \sqrt{\frac{4}{3}} (z + z^{−1})

Now consider α = (0, \frac{1}{2}, 0, −\frac{1}{2}, 0, \frac{1}{2}, 0, −\frac{1}{2}, …). If d̃_{m}(α, T_{C_{0}}) were defined as sum from k = 0 to p − 1, it would be zero for all m, but d_{m}(α, T_{C_{0}}) is not small. □

The problem, as this example shows, is that for sequences in T_{J_{0}}, (α_{0}, …, α_{p−2}) does not determine α_{p−1}. But by periodicity, α_{0}, …, α_{p−1} determines α_{p}. Thus, if we define

d̃_{m}(α, α′) := ∑_{k=0}^{p} |α_{m+k} − α′_{m+k}|  (4.21)

then

|α_{m} + p − α_{m}| ≤ d̃_{m}(α, T_{J_{0}})

Plugging this into the proofs of Lemma 3.6 and Proposition 3.5 leads quickly to

Proposition 4.6. Let C_{0} be a fixed periodic CMV matrix, then

e^{−p}∥d̃_{m}(α, T_{C_{0}})∥_{ℓ^{q}}≤ ∥d_{m}(α, T_{J_{0}})∥_{ℓ^{q}}≤ C∥d̃_{m}(α, T_{C_{0}})∥_{ℓ^{q}}  (4.22)

for any sequence of Verblunsky coefficients {α_{n}}.

5. The Magic Formula for Schrödinger Operators

In this section, we want to illustrate the potential applicability of our central idea to the spectral theory of one-dimensional Schrödinger operators,

H = −\frac{d^{2}}{dx} + V (x)  (5.1)

However, we will not pursue the applications in this paper but defer them to a future publication.

We will suppose V ∈ L_{unif}^{1}, that is, sup_{x} ∫_{x−1}^{x+1}|V (y)| dy < ∞. In that case, V is a form bounded perturbation of −\frac{d^{2}}{dx^{2}} on L^{2}(ℝ, dx) with relative bound zero, so H is a self-adjoint operator. Its form domain is the Sobolev space H^{1}(ℝ).

We need to say something about periodic Schrödinger operators. Suppose V_{0} has period L, that is,

V_{0}(x + L) = V_{0}(x)  (5.2)

For arbitrary V in L_{loc}^{1} and E ∈ ℂ, let u_{D}(x, E; V) and u_{N}(x, E; V) (we will often drop the V if it is clear which V is intended) be the solutions of

−u′′ + V u = Eu  (5.3)

obeying the boundary conditions

u_{D}(0) = 0  u′_{D}(0) = 1  u_{N}(0) = 1  u′_{N}(0) = 0  (5.4)

There is a unique solution of (5.3) in distributional sense which is absolutely continuous.

The transfer matrix that updates solutions of (5.3) (with data written as \binom{u}{u′}) is

T (x, E; V) = (\begin{matrix}u_{N}(x, E) & u_{D}(x, E) \\ u′_{N}(x, E) & u′_{D}(x, E) \end{matrix})  (5.5)

det(T) = 1 by constancy of the Wronskian. For periodic V_{0}, we define the discriminant by

Δ_{V_{0}}(E) = Tr(T (L, E; V_{0}))
= u_{N}(L, E) + u′_{D}(L, E)  (5.6)

As in the OPRL and OPUC cases, it is easy to see for the whole-line operator that

σ(−\frac{d^{2}}{dx^{2}} + V_{0}) = Δ_{V_{0}}^{−1}([−2, 2])  (5.7)

and is purely absolutely continuous. Moreover (see, e.g., [89]), if

(S_{y}u)(x) = u(x − y)  (5.8)

then H = −\frac{d^{2}}{dx^{2}} +V_{0} commutes with S_{L} and so has a direct integral decomposition,

H = ∫ ^{⊕} H(θ) \frac{dθ}{2π}  (5.9)

whose fibers, H(θ), are the operator (5.1) on [0, L] with

u(L) = e^{iθ}u(0)  u′(L) = e^{iθ}u′(0)  (5.10)

boundary conditions. H(θ) has purely discrete spectrum (i.e., (H(θ) + i)^{−1} is compact); the eigenvalues are precisely the solutions of

Δ(E) = 2 cos(θ)  (5.11)

Two periodic potentials of period L are called isospectral if and only if they have the same Δ. As in the Jacobi and CMV cases, the spectrum determines Δ, but this is more difficult to prove in the Schrödinger case. It is also known ([75] for nice V_{0}’s; [15, 39, 54, 88]) that the set of V ’s isospectral to V_{0} is a torus of dimension equal to the number of gaps which is typically infinite, so we will refer to an isospectral torus, T_{V_{0}}. We can now state the main result in this section:

Theorem 5.1. Let V_{0} be periodic obeying (5.2) and let Δ_{V_{0}} be its discriminant and T_{V_{0}} its isospectral torus. Let V be in L_{loc,unif}^{1} on ℝ and H = −\frac{d^{2}}{dx^{2}} + V. Then

Δ_{V_{0}}(H) = S_{L} + S_{−L} ⟺ V ∈ T_{V_{0}}  (5.12)

Here S_{±L} denotes the shift operator, as in (5.8).

Remarks. 1. Δ_{V_{0}}(H) is defined by the functional calculus.

2. As in the last two sections, we will provide our initial proof that

Δ_{V_{0}}(H) = S_{L} + S_{−L} ⇒ V periodic  (5.13)

and then a simpler proof using an analog of Naĭman’s lemma. This argument does not require Theorems 5.2 and 5.3 and the considerable machinery their proofs entail. That said, to show Δ_{V_{0}}(H) = S_{L} + S_{−L} plus V periodic implies V ∈ T_{V_{0}} does require Theorem 5.3, but it should be noted that one can prove Theorem 5.3 fairly easily without needing transformation formulae of Delsarte, Levitan, Gel’fand, Marchenko type.

We need two preliminaries whose proofs we defer to later in the section. We first make a definition:

Definition. For any y > 0, R_{y} consists of operators on L^{2}(ℝ) of the form

(Af)(x) = \frac{1}{2} f(x + y) + \frac{1}{2} f(x − y) + ∫_{x−y}^{x+y} K(x, z)f(z) dz  (5.14)

where K is continuous and uniformly bounded on {(x, z) : |x − z| ≤ y}.

Note. It can happen that K(x, x ± y) ≠ 0, so if we think of K as an integral kernel on ℝ × ℝ, it can be discontinuous at |x − z| = y.

Theorem 5.2. If V_{0} is L-periodic and V in L_{loc,unif}^{1}, then \frac{1}{2}Δ(H) ∈ R_{L} and

K(x, x + L) = −\frac{1}{4} ∫_{x}^{x+L}(V (z) − V_{0}(z)) dz  (5.15)

Note that (5.15) describes the ‘matrix elements’ of Δ_{V_{0}}(H) − (S_{L} + S_{−L}) that are farthest from the diagonal. Indeed, just as in the other cases, one does not need the full statement Δ_{V_{0}}(H) = S_{L} + S_{−L} to see that V is periodic, only that ⟨f, (Δ(H) − S_{L} − S_{−L})g⟩ = 0 for f supported near x_{0} and g near x_{0} + L (for all x_{0}).

Theorem 5.3. Δ(E) is an entire function which obeys

(i)  |Δ(E)| ≤ C exp(L\sqrt{|E|})  (5.16)
(ii)  lim_{\begin{matrix}E→−∞ \\ E real \end{matrix}} \frac{Δ(E)}{exp(L\sqrt{|E|})} = 1  (5.17)

Proof of Theorem 5.1. If V ∈ T_{V_{0}}, then Δ_{V} = Δ_{V_{0}}, so for the ⇐ direction we need only prove

Δ_{V_{0}}(\frac{d^{2}}{dx^{2}} + V_{0}) = S_{L} + S_{−L}  (5.18)

As before, this is equivalent to Δ_{V_{0}}(H(θ)) = 2 cos θ which follows from (5.11).

Conversely, if Δ_{V_{0}}(H) = S_{L} + S_{−L}, then from Theorem 5.2 and the periodicity of V_{0}, we see

∫_{x}^{x+L} V (z) dz = constant  (5.19)

This implies that V (x + L) − V (x) = 0 for a.e. x, that is, V is periodic.

If H(θ) are the fibers of H in the direct integral decomposition, Δ_{V_{0}}(H) = S_{L} + S_{−L} implies

Δ_{V_{0}}(H(θ)) = 2 cos θ  (5.20)

so, if Δ is the discriminant for V, we have Δ(z) = ±2 ⇒ Δ_{V_{0}}(z) = ±2. Moreover, (5.20) implies σ(H) ⊆ σ(−\frac{d^{2}}{dx^{2}} + V_{0}), so any double zero of Δ ± 2 is a double zero of Δ_{V_{0}} ± 2. It follows that

g(z) = \frac{Δ_{V_{0}}^{2}(z) − 4}{Δ^{2}(z) − 4}  (5.21)

is analytic.

Since Δ_{V_{0}} and Δ are entire functions of order \frac{1}{2} (by Theorem 5.3), g(z) is of the form

g(z) = C ∏_{j=1}^{J}(1 − \frac{z}{z_{j}} )  (5.22)

where z_{1} < z_{2} < ⋯ are bounded from below. By (5.17), lim_{E→−∞} g(E) = 1, which implies g ≡ 1, that is, Δ = Δ_{V_{0}}. □

The argument used at the end of the proof to conclude that missing zeros cannot occur is reminiscent of ideas connected with the Hochstadt–Lieberman [52] and related theorems [45, 46].

We now turn to the proofs of Theorems 5.2 and 5.3. A critical role is played by the wave equation and the transformation operator formalism of Gel’fand–Levitan, further important work is due to Delsarte, Levin, and Marchenko; see the book of Marchenko [73] for references and history.

Define for s > 0,

C_{s}(z) = cos(s\sqrt{z})  S_{s}(z) = z^{−1/2} sin(s\sqrt{z})  (5.23)

which are entire functions of z bounded on (a, ∞) for any a ∈ ℝ. Thus C_{s}(H) and S_{s}(H) are bounded operators for any H that is bounded from below. We will need to study the form of C_{s}(−\frac{d^{2}}{dx^{2}} + V). For bounded continuous V, this is discussed in Marchenko [73]. While his proofs extend to the L_{loc}^{1} case, it seems simpler to sketch the ideas:

Proposition 5.4. C_{s}^{0} := C_{s}(−\frac{d^{2}}{dx^{2}} ) ∈ R_{s}; indeed,

(C_{s}^{0}f)(x) = \frac{1}{2} [f(x + s) + f(x − s)]  (5.24)

If S_{s}^{0} := S_{s}(−\frac{d^{2}}{dx^{2}} ), then

(S_{s}^{0}f)(x) = \frac{1}{2} ∫_{x−s}^{x+s} f(y) dy  (5.25)

Remark. If w(x, s) := (C_{s}^{0}f)(x) + (S_{s}g)(x), then w obeys the wave equation (\frac{∂^{2}}{∂s^{2}} − \frac{∂^{2}}{∂x^{2}} )w = 0 with initial data w(x, 0) = f and ∂_{s}w(x, 0) = g(x). Thus the proposition basically encodes d’Alembert’s solution of the wave equation. From this point of view, Theorem 5.2 is connected to finite propagation speed for the wave equation.

Proof. Since cos is even,

cos(s|k|) = cos(sk) = \frac{1}{2} (e^{iks} + e^{−iks})  (5.26)

(5.24) is just the Fourier transform of this. (5.25) follows from

S_{s}(z) = ∫_{0}^{s} C_{t}(z) dt  (5.27)

and (5.24). □

We are heading towards

Theorem 5.5. Let V ∈ L_{loc,unif}^{1}(ℝ) and let H = −\frac{d^{2}}{dx^{2}} + V. Then C_{s}(H) ∈ R_{s} and the associated kernel K_{s} of (5.14) obeys

K_{s}(x, x + s) = −\frac{1}{4} ∫_{x}^{x+s} V (u) du  (5.28)

and for each t ∈ (0, ∞),

sup_{x,y,|s|≤t} |K_{s}(x, y)| < ∞  (5.29)

In addition,

(S_{s}(H)f)(x) = ∫_{x−s}^{x+s} L_{s}(x, y)f(y) dy  (5.30)

where

L_{s}(x, x + s) = \frac{1}{2}  (5.31)

Lemma 5.6. It suffices to prove Theorem 5.5 for s small.

Proof. Since cos(2u) = 2 cos^{2}(u) − 1, one sees

C_{2s}(A) = 2C_{s}(A)^{2} − 1  (5.32)

Thus, if C_{s} ∈ R_{s}, one sees C_{2s} ∈ R_{2s} and

K_{2s}(x, y) = K_{s}(x, y + s) + K_{s}(x, y − s) + K_{s}(x + s, y) + K_{s}(x − s, y) (5.33)

where K(x, y) = 0 if |x − y| > s. Thus

K_{2s}(x, x + 2s) = K_{s}(x, y + s) + K_{s}(x + s, y + 2s)  (5.34)

This shows that if the formula is known for |s| ≤ T, one gets it successively for 2T, 4T, 8T, . . ..

Using (5.27), one sees that the result for C_{s}(H) implies (5.30) and (5.31). □

Proof of Theorem 5.5. If A is a bounded self-adjoint operator on H which is bounded from below, and B is the operator on H ⊕ H given by

B = ( \begin{matrix}0 & 1 \\ −A & 0 \end{matrix} )  (5.35)

then

e^{sB} = ( \begin{matrix}C_{t}(A) & S_{t}(A) \\ −AS_{t}(A) & C_{t}(A) \end{matrix} )  (5.36)

This formula can be checked by showing that the right side of (5.36) is a bounded semigroup whose derivative at t = 0 is B. DuHamel’s formula for A, Ã bounded says that

e^{tB̃} = e^{tB} + ∫_{0}^{t} e^{sB}(B̃ − B)e^{(t−s)B̃} ds  (5.37)
= e^{tB} + ∫_{0}^{t} e^{sB̃}(B̃ − B)e^{(t−s)B} ds  (5.38)

Using (5.36), we obtain

C_{t}(Ã) = C_{t}(A) − ∫_{0}^{t} S_{s}(Ã)(Ã − A)C_{t−s}(A) ds  (5.39)
= C_{t}(A) − ∫_{0}^{t} S_{s}(A)(Ã − A)C_{t−s}(Ã) ds  (5.40)

By taking limits, it is easy to obtain these formulae for A = −\frac{d^{2}}{dx^{2}} , Ã = −\frac{d^{2}}{dx^{2}} +V with V bounded. By obtaining a priori bounds below depending only on certain L^{1} norms of V, we get estimates for V in L^{1} and so, using the lemma, prove the theorem.

By iterating (5.40), one gets an expansion (which converges if V is bounded and whose estimates then extend),

C_{t}(−\frac{d^{2}}{dx^{2}} + V (x)) = C_{t}^{(0)} + ∑_{n=1}^{∞} C_{t}^{(n)}  (5.41)

C_{t}^{(n)} = (−1)^{n} ∫_{0≤s_{1}+···+s_{n}≤t} S_{s_{1}}^{(0)}V S_{s_{2}}^{(0)} … V S_{s_{n}}^{(0)}V C_{t−s_{1}−···−s_{n}}^{(0)} ds_{1}, … ds_{n} (5.42)

Apply the integrand in C_{t}^{(n)} to a function f and evaluate at x for fixed s_{1}, …, s_{n}. Each S_{s_{j}}^{(0)}V evaluates V at points and integrals using (5.25). The integrands in V are in the interval (x − t, x + t), so if we take absolute values, we see this integrand is bounded by

(\frac{1}{2} ∫_{x−t}^{x+t}|V (y)| dy)^{n}[\frac{1}{2} f(x + t − s_{1} − ⋯ − s_{n}) + \frac{1}{2} f(x − t + s_{1} + ⋯ + s_{n})]

Now we can do the integral over s_{1}, …, s_{n}. For t − s_{1} − ⋯ − s_{n} fixed, the new integrand is independent of s_{1}, …, s_{n−1} and is bounded by t^{n−1}. We find

|(C_{t}^{(n)}f)(x)| ≤ t^{n−1}(\frac{1}{2} ∫_{x−t}^{x+t}|V (y)| dy)^{n} ∫_{x−t}^{x+t}|f(y)| dy  (5.43)

Moreover, C_{t}^{(n)} has a continuous integral kernel K_{t}^{(n)}(x, y) supported in |x − y| ≤ t. Since V is uniformly locally L^{1}, by taking t small, we can be sure sup_{x} \frac{1}{2} ∫_{x−t}^{x+t}|V (y)| dy < 1, which yields uniform convergence of K_{t}^{(n)} to a uniformly bounded kernel.

By (5.27), we get (5.30) from C_{s}(H) ∈ R_{s}, and (5.31) comes from noting that

|L_{s}(x, y) − \frac{1}{2}| ≤ (s − |x − y|) sup_{x,y,u≤s} |K_{u}(x, y)|  (5.44)

Finally, using (5.39), we see that

K_{t}(x, x + t) = −\frac{1}{2} ∫_{0}^{t} L_{s}(x, x + s)V (x + s) ds

proving (5.28). □

To complete the proofs of Theorems 5.2 and 5.3 (and so Theorem 5.1), we need the transformation formulae of Delsarte, Levitan, Gel’fand, and Marchenko [73]:

Theorem 5.7. If V ∈ L^{1}([0, R]) for R < ∞, then there exist functions K_{N}, K_{D} C^{1} in {(y, x) : 0 ≤ y ≤ x ≤ R} so that for 0 ≤ x ≤ R,

u_{N}(x, E) = C_{x}(E) + ∫_{0}^{x} K_{N}(y, x)C_{y}(E) dy  (5.45)
u_{D}(x, E) = S_{x}(E) + ∫_{0}^{x} K_{D}(y, x)S_{y}(E) dy  (5.46)

Moreover,

K_{D}(x, x) = K_{N}(x, x) = \frac{1}{2} ∫_{0}^{x} V (t) dt  (5.47)

Remarks. 1. These formulae are in Marchenko [73, p. 9 and (1.2.28)]. He supposes V is continuous, but his proof works if V is L^{1}; indeed, see Remark 2.

2. Defining ũ_{X}(x, k) = u_{X}(x, k^{2}) for X = D, N and Q_{X}(x, y) as the Fourier transform of ũ_{X} in k, we see (5.3) becomes

\frac{∂^{2}Q}{∂x^{2}} − \frac{∂^{2}Q}{∂y^{2}} = V Q(x, y)  (5.48)

with initial conditions

Q_{N}(x = 0, y) = δ(y)  Q′_{n}(x = 0, y) = 0
Q_{D}(x = 0, y) = 0  Q′_{D}(x = 0, y) = δ(y)

Thus, Theorem 5.7 is essentially Theorem 5.5 with a time-dependent V used. 3. By (5.6), (5.45), and (5.46), we obtain a critical representation for Δ:

\begin{aligned}Δ(E) = 2C_{x}(E) & + ∫_{0}^{L} L_{1}(t)C_{t}(E) dt \\ & + ∫_{0}^{L} L_{2}(t)S_{t}(E) dt + K_{D}(L, L)S_{L}(E) \end{aligned}  (5.49)

where L_{1}, L_{2} are continuous in [0, L]. Indeed,

L_{1}(t) = K_{N}(t, L) L_{2}(t) = \frac{∂}{∂x} K_{D}(t, x)|_{x=L}

Proof of Theorem 5.3. The analyticity is immediate from (5.49) as is (5.16) given

|C_{x}(E)| + |S_{x}(E)| ≤ C exp(x\sqrt{|E|})

Moreover, since for t < L,

lim_{E→−∞} \frac{C_{t}(E)}{C_{L}(E)} = 0  and  lim_{E→−∞} \frac{S_{L}(E)}{C_{L}(E)} = 0

we have (5.17). □

Proof of Theorem 5.2. By (5.49) and Theorem 5.5, \frac{1}{2}Δ(−\frac{d^{2}}{dx^{2}} + V) is in R_{L}. Moreover, the only terms contributing to K(x, x + L) come from C_{L}(−\frac{d^{2}}{dx^{2}} + V) and K_{D}(L, L)S_{L}(−\frac{d^{2}}{dx^{2}} + V). By (5.28), (5.31), and (5.47),

K(x, x + L) = −\frac{1}{4} ∫_{x}^{x+L} V (y) dy + \frac{1}{2}(\frac{1}{2} ∫_{0}^{L} V_{0}(y) dy)

which, given the periodicity of V_{0}, is (5.15). □

There is a second proof of (5.13). It depends on this analog of Naĭman’s lemma:

Lemma 5.8. If V is L_{loc,unif}^{1} and −\frac{d^{2}}{dx^{2}} + V commutes with S_{L} + S_{−L}, then

V (x + L) = V (x)  (5.50)

Proof. Suppose first that V is bounded. Then S_{L} + S_{−L} leaves D(−\frac{d^{2}}{dx^{2}} ) invariant and commutes with it, so S_{L} + S_{−L} commutes with V. If f is supported in a small neighborhood of x_{0}, (x_{0} − δ, x_{0} + δ) with |δ| < L/2, then (S_{L} + S_{−L})(V f) is two separate pieces V (x − L)f(x − L) supported near x_{0} + L and V (x + L)f(x + L) supported near x_{0} − L, while V (S_{L} + S_{−L})f is two pieces V (x)f(x − L) and V (x)f(x + L). Since the pieces are disjoint,

V (x)f(x − L) = V (x + L)f(x − L)

which implies (5.50).

For general V, take g ∈ C_{0}^{∞}(ℝ) with ∫ g(x) dx = 1 and note that ∫ g(x)S_{x}(−\frac{d^{2}}{dx^{2}} + V)S_{−x} dx is −\frac{d^{2}}{dx^{2}} +g∗V and it commutes with S_{L}+S_{−L} also. But g∗V is bounded, so it is periodic. (5.50) follows by using an approximate δ-function. □

6. Block Jacobi Matrices and Matrix Orthogonal Polynomials

What the magic formula suggests is that the Jacobi matrix J has parameters that approach an isospectral torus if and only if Δ(J) approaches S^{p} + S^{−p}. Δ(J) is a matrix of width 2p+1 (i.e., Δ(J)_{kℓ} = 0 if k−ℓ ∉ {0, ±1, …, ±p}) and S^{p} +S^{−p} is a matrix with 1’s at the extremes.

A matrix of width 2ℓ + 1 has the structure of a tridiagonal matrix if rewritten in terms of ℓ × ℓ blocks and S^{ℓ} + S^{−ℓ} corresponds to B_{k} = 0, A_{k} = 1, the identity matrix, so Δ(J) ∼ S^{ℓ} + S^{−ℓ}, at the matrix level, approaches the ‘free case.’ This will allow us to reduce our main theorems to matrix analogs of the theorems on perturbations of the free case.

Of course, the association of the block matrix to orthogonal polynomials is critical—the orthogonality will be with respect to a matrix-valued measure. There is a huge literature on MOPRL (see, e.g., [16, 23, 31, 32, 33, 34, 35, 36, 37, 38, 55, 71, 72, 112, 113]) and MOPUC (see, e.g., [8, 24, 25, 26, 27, 28, 29, 42, 68, 70, 90, 107, 114]). In this section, our main purpose is to set notation and discuss the important notion of equivalent families of block Jacobi matrices, a notion discussed more explicitly in [20].

Given a semi-infinite complex matrix M = {m_{ij}}_{1≤i,j≤∞} and ℓ = 1, 2, …, we define the ℓ × ℓ block decomposition as the family of ℓ × ℓ matrices {M_{pq}}_{1≤p,q≤∞} by

(M_{pq})_{ij} = m_{ℓ(p−1)+i, ℓ(q−1)+j}  i, j = 1, … ℓ  (6.1)

Definition. A block Jacobi matrix is an M where

M_{pq} = \begin{cases}B_{p} & if q = p ≥ 1 \\ A_{p} & if q = p + 1, p ≥ 1 \\ A_{p−1}^{∗} & if q = p − 1, p ≥ 2 \\ 0 & |p − q| ≥ 2 \end{cases}  (6.2)

with each A_{p} invertible and each B_{p} Hermitian.

We will start using J for such matrices.

In analogy with the scalar case, one may be tempted to require

A_{p} > 0  (6.3)

but to include Δ(J), we do not want to do that exclusively. If (6.3) holds, we say that J is of type 1.

If instead

A_{1} … A_{n} > 0  (6.4)

for all n, we say J is of type 2.

An ℓ×ℓ matrix, K, is said to be in L if it is lower triangular with strictly positive diagonal elements, that is,

K_{ij} = \begin{cases}0 & if i < j \\ > 0 & if i = j \end{cases}  (6.5)

If each A_{p} ∈ L, we say that J is of type 3.

The calculations in Section 3 and 4 show:

Proposition 6.1. (i) If Δ is the discriminant of a periodic Jacobi matrix, J_{0}, of period ℓ, then for any Jacobi matrix, J, Δ(J) = J is a block Jacobi matrix of type 3.

(ii) If Δ is the discriminant of a periodic CMV matrix, C_{0}, of even period ℓ, then for any CMV matrix, C, of Δ(C) = J is a block Jacobi matrix of type 3.

We will see that distinct J ’s may correspond to the same measure. Indeed, in the scalar case, the b_{n}’s and |a_{n}|’s are fixed by the measure, but the arg(a_{n})’s are arbitrary. Thus, we define

Definition. Two block Jacobi matrices, J and J̃, are called equivalent if and only if there is an ℓ × ℓ block diagonal unitary U = 1 ⊕ U_{2} ⊕ U_{3} ⊕ ⋯ (we will use U_{1} for 1) so that

J̃ = UJ U^{−1}  (6.6)

This is equivalent to

B̃_{n} = U_{n}B_{n}U_{n}^{−1}  (6.7a)
Ã_{n} = U_{n}A_{n}U_{n+1}^{−1}  (6.7b)

We will be interested in equivalence classes of J ’s.

Proposition 6.2 ([20]). Each equivalence class of J ’s has exactly one element each of type 1, type 2, and type 3.

Definition. The Nevai class is the set of J ’s for which

B_{n} → 0  A_{n}^{∗}A_{n} → 1  (6.8)

The following is immediate from (6.7):

Proposition 6.3. If some J is in the Nevai class, so are all equivalent J ’s.

Damanik, Pushnitski, and Simon [20] prove that

Proposition 6.4 ([20]). If J is in the Nevai class and is type 1, type 2, or type 3, then

A_{n} → 1  (6.9)

We will sometimes need the MOPRL, the matrix orthogonal polynomials. What we describe here are the left OPs. There are also right OPs (see [20]), which we do not need here. An ℓ-dimensional matrix-valued measure is a positive scalar measure dρ_{t}(x) and a nonnegative ℓ × ℓ matrix-valued function M(x). The matrix-valued measure

dρ(x) = M(x) dρ_{t}(x)  (6.10)

can always be normalized by

Tr(M(x)) = ℓ  (6.11)

We will always assume dρ is normalized, that is,

∫ dρ(x) = 1  (6.12)

The proper notion of nontriviality is a little subtle; it is discussed in detail in [20]. For our purpose here, it is sufficient that ⟨⟨·, ·⟩⟩_{L} defined below is nondegenerate on polynomials.

If f, g are two ℓ-dimensional matrix-valued functions, we define

⟨⟨f, g⟩⟩_{L} = ∫ g(x)M(x)f(x)^{†} dρ_{t}(x)

where, following [98, Section 2.13], we use ^{†} for Hermitian adjoint (so for MOPUC, one does not confuse Szegő dual Φ_{n}^{∗} with adjoints). The name ‘left’ comes from the fact that if C is an ℓ × ℓ matrix, then

⟨⟨f, Cg⟩⟩_{L} = C⟨⟨f, g⟩⟩_{L}  (6.13)
⟨⟨Cf, g⟩⟩_{L} = ⟨⟨f, g⟩⟩_{L}C^{†}  (6.14)

We will normally just write ⟨⟨·, ·⟩⟩ from now on.

Left orthonormal polynomials are of the form

p_{n}(x) = κ_{n}x^{n} + lower order

with matrix coefficients, defined by

⟨⟨p_{n}, p_{m}⟩⟩ = δ_{nm}1  (6.15)

So long as dρ is nontrivial, the p_{n} exist. They are not unique since if {U_{n}}_{n=1}^{∞} are unitary ℓ × ℓ matrices,

p̃_{n}(x) = U_{n+1}p_{n}(x)  (6.16)

are also MOPRL. We demand κ_{0} = 1, that is, p_{0}(x) = 1, and so U_{1} = 1.

{p_{j}}_{j=0}^{n} are a left module basis for matrix polynomials of degree n, that is, if f is any polynomial of degree n, then there are unique ℓ × ℓ matrices f_{0}, …, f_{n} so that

f(x) = ∑_{j=0}^{n} f_{j}p_{j}(x)

Indeed,

f_{j} = ⟨⟨p_{j}, f⟩⟩  (6.17)

For n = 1, 2, …, define

B_{n} = ⟨⟨p_{n−1}, xp_{n−1}⟩⟩  A_{n} = ⟨⟨p_{n}, xp_{n−1}⟩⟩  (6.18)

Then, since xp_{j} = ∑_{ℓ=0}^{j+1} C_{ℓ}p_{ℓ} implies ⟨⟨p_{j}, xp_{n}⟩⟩ = ⟨⟨xp_{j}, p_{n}⟩⟩ = 0 if j ≤ n − 2, we have

xp_{n}(x) = A_{n+1}p_{n+1}(x) + B_{n+1}p_{n}(x) + A_{n}^{∗}p_{n−1}(x)  (6.19)

This implies A_{n+1}κ_{n+1} = κ_{n} so

κ_{n} = (A_{1} … A_{n})^{−1}  (6.20)

and the type 2 condition is equivalent to κ_{n} > 0.

Looking at (6.19), we see that (6.16) holds for p̃_{n}, p_{n} if and only if Ã_{n}, B̃_{n} are related to A_{n}, B_{n} by (6.7). Jacobi matrix equivalence is just a ‘change of phase’ in the MOPRL.

Given a block Jacobi matrix, we can view it as acting on the Hilbert space ℓ^{2}({1, 2, … }, ℂ^{ℓ}) with inner product

⟨f, g⟩ = ∑_{n=1}^{∞}⟨f_{n}, g_{n}⟩_{ℂ^{ℓ}}  (6.21)

If {e_{j}}_{j=1}^{ℓ} is the standard basis of ℂ^{ℓ}, then {δ_{k;j}}_{k=1}^{∞}_{j=1}^{ℓ}, defined by

(δ_{k;j})_{n} = δ_{kn}e_{j}  (6.22)

is a basis. J acts on ℓ^{2}({1, 2, … }, ℂ^{ℓ}) via

(J f)_{n} = A_{n−1}^{∗}f_{n−1} + B_{n}f_{n} + A_{n}f_{n+1}  (6.23)

(with A_{0} = 0).

The spectral measure for J is the ℓ × ℓ matrix-valued measure with

⟨δ_{0;j}, f(J)δ_{0;k}⟩ = ∫ f(x) dρ_{jk}(x)  (6.24)

for any scalar-valued function f. It is easy to see (e.g., [20]) that this map from J to ρ inverts the one given by forming the MOPRL and defining J by (6.18). Moreover, J and J̃ are equivalent if and only if dρ̃ ≡ dρ.

The m-function is defined by

m(E) = ∫ \frac{1}{x − E} dρ(x)  (6.25)
= ⟨δ_{0; ·} , (J − E)^{−1}δ_{0; ·} ⟩  (6.26)

It is an ℓ × ℓ matrix-valued Herglotz function:

Im E > 0 ⇒ Im m(E) > 0  (6.27)

that is, \frac{1}{2i}(m − m^{†}) is positive definite in the upper half-plane. For information on matrix Herglotz functions, see [3, 4, 11, 12, 40, 44, 47, 48, 51, 56, 62, 63, 92, 110]. Obviously, by (6.25), m is constant over equivalence classes.

As in the scalar case (see [98, Section 1.2]), one has that for a.e. x ∈ ℝ, lim_{ε↓0} m(x + iε) ≡ m(x + i0) exists and

dρ_{ac}(x) = π^{−1} Im m(x + i0) dx  (6.28)

Here

dρ_{ac}(x) = M(x) dρ_{t;ac}(x)  (6.29)

where dρ_{t;ac} is the a.c. part of dρ_{t}. Alternatively, dρ_{ac} is the unique matrix-valued measure which is a.c. (i.e., ρ_{ac}(I) = 0 for any set with |I| = 0) and where there is a set K with |K| = 0 so (ρ − ρ_{ac})(ℝ \ K) = 0.

Given a block Jacobi matrix, J, by J^{(n)} we mean the matrix with the top n (block matrix) rows and leftmost n columns removed, that is,

B_{k}^{(n)} = B_{k+n}  A_{k}^{(n)} = A_{k+n}  (6.30)

We write m^{(n)}(z) for the m-function associated to J^{(n)}. Equivalent J ’s do not have the same m^{(n)} for n ≥ 1 (although they are unitarily related). We see m^{(0)} ≡ m.

We will need the following result of Aptekarev–Nikishin [6] (see also [20]), a matrix analog of the well-known Jacobi–Stieltjes recursion for OPRL:

Theorem 6.5 ([6, 20]). We have that

m^{(n)}(E)^{−1} = E − B_{n+1} − A_{n+1}m^{(n+1)}(E)A_{n+1}^{∗}  (6.31)

for n = 0, 1, 2, . . ..

Next, we need to note the following analog of a well-known scalar result (see, e.g., [100]) proven in [20]:

Theorem 6.6. Let J be a block Jacobi matrix with σ_{ess}(J) ⊂ [a, b]. Then, for any ε, there is a K so that for k ≥ K,

σ(J^{(k)}) ⊂ [a − ε, b + ε]  (6.32)

Finally, we need to look at poles and zeros of det(m(z)). In the scalar (ℓ = 1) case, poles occur precisely at eigenvalues of J and zeros at eigenvalues of J^{(1)}, the once stripped Jacobi matrix. In that scalar case, these eigenvalues are distinct.

In the matrix case, J and J^{(1)} can have eigenvalues in common (as can be easily arranged by taking a direct sum of suitable scalar J’s) so there can be cancellations. We say a scalar meromorphic function, f(z), has a zero/pole of order k ∈ ℤ at z_{0} ∈ ℂ if (z − z_{0})^{−k}f(z) has a finite nonzero limit as z → z_{0}. We will need the following result from [20]:

Theorem 6.7 ([20]). Let x_{0} ∈ ℝ. Let q_{0} be the multiplicity of x_{0} as an eigenvalue of J, and q_{1} its multiplicity as an eigenvalue of J_{1}. Then

(a) q_{0} + q_{1} ≤ ℓ

(b) det(m(z)) has a zero/pole of order q_{1} − q_{0}.

We will also need the following result from Aptekarev–Nikishin [6]:

Theorem 6.8. Let J be a block Jacobi matrix with σ_{ess}(J) = [−2, 2], σ(J) \ σ_{ess}(J) a finite set and with g(x) = dρ_{ac}(x)/dx we have

∫ (4 − x^{2})^{−1/2} log(det(g(x)) dx > −∞  (6.33)

Suppose J is type 2. Then

lim_{n→∞} A_{1} … A_{n}

exists and is a strictly positive matrix.

The authors of [20] hope to extend this result to allow point spectrum obeying ∑_{j}(|E_{j}| − 2)^{1/2} < ∞ before [20] appears.

7. A Denisov–Rakhmanov Theorem for MOPRL

As preparation for proving Theorem 1.2 in Section 8, in this section we will prove

Theorem 7.1. Let dρ be a nondegenerate ℓ × ℓ matrix-valued measure on ℝ with associated block Jacobi matrix J of type 3 so that

(i)  σ_{ess}(J) = [−2, 2]  (7.1)
(ii)  dρ = f(x) dx + dρ_{s}  (7.2)

with dρ_{s} singular and

det(f(x)) > 0  (7.3)

a.e. on [−2, 2]. Then

B_{n} → 0  A_{n} → 1  (7.4)

Remark. (7.3) says the a.c. spectrum has multiplicity ℓ.

If (7.1) is replaced by the stronger σ(J) = [−2, 2] and type 3 by type 2, this is a theorem of Yakhlef–Marcellán [112]. We will prove Theorem 7.1 by modifying their proof.

The shift from type 2 to 3 is easy on account of Proposition 6.4. By applying the argument of [112], we get Ã_{n} → 1 for the equivalent J̃ of type 2, conclude the whole equivalence class is in the Nevai class, and see A_{n} → 1. So we will only worry about the changes needed to go from σ(J) = [−2, 2] to σ_{ess}(J) = [−2, 2], where we follow Denisov’s approach for the scalar case [30].

[112] relies on a matrix version of Rakhmanov’s theorem proven by van Assche [107]. We need to extend it slightly to allow a.c. spectrum on a large subset of ∂𝔻 rather than all of ∂𝔻:

Theorem 7.2. Let dμ be an ℓ × ℓ matrix-valued measure on ∂𝔻 and let {α_{n}}_{n=0}^{∞} denote its matrix Verblunsky coefficients . Suppose

dμ = w(θ) \frac{dθ}{2w} + dμ_{s}  (7.5)

where dμ_{s} is singular, and let

Ω = {θ : det(w(θ)) > 0}  (7.6)

Then

lim sup_{n→∞} ∥α_{n}∥ ≤ 2\sqrt{2ℓ} (1 − (\frac{|Ω|}{2π})^{3})^{1/2}  (7.7)

Remarks. 1. For notation on MOPUC, see [20].

2. Where we use {α_{n}}_{n=0}^{∞}, van Assche [107] uses {H_{n}}_{n=1}^{∞} related to α_{n} by

H_{n} = −α_{n−1}^{†}  (7.8)

3. We follow notation from [107] and the variant of the scalar proof as in [99, Section 9.1] where a_{n}, b_{n}, c_{n}, d_{n} below all appear.

We define

a_{n} = ∥α_{n}∥
b_{n,q} = \frac{1}{2π} ∫_{0}^{2π} ∥[φ_{n}^{L}(e^{iθ})φ_{n+q}^{L}(e^{iθ})^{−1}] [φ_{n}^{L}(e^{iθ})φ_{n+q}^{L}(e^{iθ})^{−1}]^{∗} − I∥ dθ
c_{n,q} = \frac{1}{2πℓ} ∫_{0}^{2π} Tr(φ_{n}^{L}(e^{iθ})φ_{n+q}^{L}(e^{iθ})^{−1}[φ_{n+q}^{L}(e^{iθ})^{∗}]^{−1}φ_{n}^{L}(e^{iθ})^{∗})^{1/2} dθ

d_{n} = \frac{1}{2πℓ} ∫_{0}^{2π} Tr([φ_{n}^{L}(e^{iθ})w(θ)φ_{n}^{L}(e^{iθ})^{∗}]^{1/2}) dθ

Proposition 7.3. For every n ≥ 0, we have that

a_{n} ≤ b_{n,q}  for every q ≥ 1  (7.9)
b_{n,q}^{2} ≤ 8ℓ^{2}(1 − c_{n,q})  for every q ≥ 1  (7.10)
d_{n}^{2} ≤ inf_{q≥1} c_{n,q}  (7.11)

Moreover, we have that

(\frac{|Ω|}{2π})^{3/2} ≤ lim inf_{n→∞} d_{n}  (7.12)

Consequently,

lim sup_{n→∞} a_{n} ≤ 2\sqrt{2} ℓ(1 − (\frac{|Ω|}{2π})^{3})^{1/2}  (7.13)

Proof. The second to last displayed formula on [107, p. 7] is (7.9). The estimates on the bottom half of [107, p. 12] show that

b_{n,q}^{2} = \frac{1}{4π^{2}} (∫_{0}^{2π} ∥[φ_{n}^{L}(e^{iθ})φ_{n+q}^{L}(e^{iθ})^{−1}] [φ_{n}^{L}(e^{iθ})φ_{n+q}^{L}(e^{iθ})^{−1}]^{∗} − I∥ dθ)^{2}

≤ \frac{2ℓ}{π} ∫_{0}^{2π} ∥([φ_{n}^{L}(e^{iθ})φ_{n+q}^{L}(e^{iθ})^{−1}] [φ_{n}^{L}(e^{iθ})φ_{n+q}^{L}(e^{iθ})^{−1}]^{∗})^{1/2} − I∥^{2} dθ

≤ \frac{2ℓ}{π} (4πℓ − 4πℓ c_{n,q})

= 8ℓ^{2}(1 − c_{n,q})

which is (7.10). The third displayed formula on [107, p. 14] is (7.11).

Now, mimicking the estimates on the bottom half of [107, p. 14],

∫_{Ω} Tr([f(θ)w(θ)f(θ)^{∗}]^{1/4}) dθ

≤ (2πℓ ∫_{Ω} Tr(f(θ)φ_{n}^{L}(e^{iθ})^{−1}(φ_{n}^{L}(e^{iθ})^{∗})^{−1}f(θ)^{∗}) \frac{dθ}{2π} )^{1/4}(2πℓ d_{n})^{1/2}

Taking n → ∞, we see that

∫_{Ω} Tr([f(θ)w(θ)f(θ)^{∗}]^{1/4}) dθ

≤ (2πℓ Tr ∫_{Ω} f(θ)dμ(θ)f(θ)^{∗})^{1/4}(2πℓ lim inf_{n→∞} d_{n})^{1/2}

Removing the singular part as in [107], we obtain

∫_{Ω} Tr([f(θ)w(θ)f(θ)^{∗}]^{1/4}) dθ

≤ (2πℓ Tr ∫_{Ω} f(θ)w(θ)f(θ)^{∗}dθ)^{1/4}(2πℓ lim inf_{n→∞} d_{n})^{1/2}

Proceeding as in [107, pp. 15–16], it then follows that

|Ω|ℓ ≤ (2πℓ |Ω|ℓ)^{1/4}(2πℓ lim inf_{n→∞} d_{n})^{1/2}

which implies (7.12).

Putting these estimates together,

a_{n} ≤ b_{n,1} ≤ 2\sqrt{2} ℓ(1 − c_{n,1})^{1/2} ≤ 2\sqrt{2} ℓ(1 − d_{n}^{2})^{1/2}

and hence

lim sup_{n→∞} a_{n} ≤ 2\sqrt{2} ℓ(1 − (\frac{|Ω|}{2π})^{3})^{1/2}

which is (7.13). □

In particular, for 2π − |Ω| small,

lim sup_{n→∞} a_{n} = O((1 − (\frac{|Ω|}{2π})^{3})^{1/2}) = O((2π − |Ω|)^{1/2})

as in the scalar case.

We have thus proven Theorem 7.2. To get Theorem 7.1, we follow [112] using the analog of Denisov’s arguments for the case ℓ = 1.

Proof of Theorem 7.1. By Proposition 6.4, we need only prove for the type 2 choice, for any ε > 0, we have

lim sup (∥Ã_{n} − 1∥ + ∥B̃_{n}∥) ≤ ε  (7.14)

By the Szegő mapping and Geronimus connection formulae in [112], this holds by Theorem 7.2 so long as for any ε′ > 0, we can find k so σ(J^{(k)}) ⊂ [−2 − ε′, 2 + ε′], and this is true by Theorem 6.6. □

8. A Denisov–Rakhmanov Theorem for Periodic OPRL

Our main goal in this section is to prove Theorem 1.2. We will also prove the ‘hard’ half of Theorem 1.1. The simplicity of the proof shows the magic in the magic formula!

Proof of Theorem 1.2. By a right limit of J, we mean a two-sided Jacobi matrix, J_{r}, (but with some a’s allowed to vanish) so that for some subsequence n_{j} → ∞ and any k ∈ ℤ,

a_{n_{j}+k} → (a_{r})_{k} b_{n_{j}+k} → (b_{r})_{k} (8.1)

By our standing convention, Jacobi parameters are uniformly bounded, so by compactness, if d_{m}((a, b), T_{J_{0}}) ↛ 0, there exists a right limit J_{r} ∉ T_{J_{0}}. Thus, it suffices to show that any right limit J_{r} has J_{r} ∈ T_{J_{0}}.

By the hypotheses of Theorem 1.2, the spectral mapping theorem, and the fact that Δ maps σ_{ess}(J_{0}) to [−2, 2] with a p-fold cover on (−2, 2), we see that

Δ(J)_{ess} = [−2, 2]

and Δ(J) has a.c. spectrum of multiplicity p. So thinking of J ≡ Δ(J) as a block Jacobi matrix, J is of type 3 and the hypotheses of Theorem 7.1 apply. It follows that A_{n} → 1, B_{n} → 0. This means that

Δ(J_{r}) = S^{p} + S^{−p}

so by the magic formula (Theorem 3.1), J_{r} ∈ T_{J_{0}}. □

Rakhmanov’s theorem is often related to issues of w-lim|p_{n}|^{2} dμ and to the density of zeros. We note that there are also results of that genre here:

Theorem 8.1. If J_{0} is a periodic Jacobi matrix of period p and J is a Jacobi matrix with bounded Jacobi parameters whose right limits all lie in T_{J_{0}} (in particular, if the hypotheses of Theorem 1.2 hold), then (with dμ the measure for J) (a)

w-lim_{n→∞} \frac{1}{p} ∑_{j=1}^{p} |p_{j+n}(x)|^{2} dμ(x) = dν  (8.2)

the density of zeros for J_{0}.

(b) The density of zeros of p_{n}(x) converges to dν.

Proof. If J_{1} ∈ T_{J_{0}}, then the spectral measure dμ_{k}^{(J_{1})} associated to δ_{k} has period p in k since J_{1} is periodic. Thus lim_{N→∞} \frac{1}{2N+1} ∑_{|j|≤N} dμ_{j}^{(J_{1})} = \frac{1}{p} ∑_{j=1}^{p} dμ_{j}^{(J_{1})}, but the limit is dν by the discussion in Subsection 2.8. Since

∫ x^{ℓ}|p_{j}^{(J)}(x)|^{2} dμ(x) = ⟨δ_{j}, J^{ℓ}δ_{j}⟩  (8.3)

and J_{j,j}, J_{j,j±1} is very close to some (J_{1})_{j,j}, (J_{1})_{j,j+1} for |j − j_{0}| ≤ M for fixed M and j_{0} → ∞, we see that moments of LHS of (8.2) are close to moments of dν. This proves (a).

If J^{(n)} denotes the top left n × n submatrix of J, then

∫ x^{ℓ} dν_{n} = \frac{1}{n} Tr((J^{(n)})^{ℓ})

so

lim_{n→∞} ∫ x^{ℓ} dν_{n} = lim_{n→∞} ∫ x^{ℓ}[ \frac{1}{n} ∑_{j=0}^{n−1} p_{j}(x)^{2} dμ(x)]

and thus (a) implies (b). □

We also have

Theorem 8.2. If d_{m}((a, b), T_{J_{0}}) → 0, then

σ_{ess}(J) ⊂ σ_{ess}(J_{0})

Proof. By the magic formula, compactness, and the fact that every right limit of J is in T_{J_{0}}, we see that every right limit of Δ(J) is S^{p} + S^{−p}, that is, A_{n} → 1, B_{n} → 0. Thus, by Weyl’s theorem, σ_{ess}(Δ(J)) = [−2, 2]. Since

σ_{ess}(Δ(J)) = Δ(σ_{ess}(J))

we see σ_{ess}(J) ⊂ Δ^{−1}([−2, 2]) = σ_{ess}(J_{0}). □

Remarks. 1. Since Δ is p to 1, we cannot conclude that σ_{ess}(J_{0}) = σ_{ess}(J) from Δ(σ_{ess}(J_{0})) = Δ(σ_{ess}(J)).

2. That σ_{ess}(J_{0}) ⊂ σ_{ess}(J) is a simple trial function argument given that J must have some right limits; see [66, 67].

9. Denisov–Rakhmanov Sets

In this section, we want to show how one can take suitable limits of Theorem 1.2 to get a ‘cheap’ proof of similar theorems in other nonperiodic cases. We will also present an insight into the proper general form of Denisov–Rakhmanov-type theorems.

The right limits we have discussed so far involve the weak product topology on the Jacobi parameters, so we will emphasize this fact by using the phrase ‘weak right limits’ in this section. We are also interested in limits in the ℓ^{∞}-topology for two-sided sequences, that is, {c_{n}^{(k)}}_{n=−∞}^{∞} → {c_{n}^{(∞)}}_{n=−∞}^{∞} in this topology if and only if, as k → ∞,

sup_{n} |c_{n}^{(k)} − c_{n}^{(∞)}| → 0

In terms of weak limits, we note the following:

Proposition 9.1. Let E be a closed set. Let J be a half-line Jacobi matrix with

Σ_{ac}(J) = σ_{ess}(J) = E  (9.1)

Let J_{r} be a weak right limit of J. Then

Σ_{ac}(J_{r}) = σ(J_{r}) = E  (9.2)

Remark. Note that (9.2) has σ(J_{r}), not merely σ_{ess}(J_{r}).

Proof. By results in [66],

σ(J_{r}) ⊂ E ⊂ Σ_{ac}(J_{r})

Since Σ_{ac}(J_{r}) ⊂ σ(J_{r}) trivially, (9.2) holds. □

Recall that a sequence {c_{n}}_{n=−∞}^{∞} is called uniformly almost periodic (in the general theory of almost periodic functions, this defines ‘almost periodic’—we add ‘uniformly’ because the term is sometimes used in a weaker sense in the spectral theory literature) if and only if {c^{(ℓ)}}_{ℓ=−∞}^{∞} given by (c^{(ℓ)})_{n} = c_{n+ℓ} has compact closure in the ℓ^{∞}-topology.

Definition. A set E is called essentially perfect if and only if E is closed, and for all x ∈ E and δ > 0, |(E − δ, E + δ)| > 0.

Remark. Essentially perfect sets are precisely the sets, E, for which there is a purely a.c. measure dρ with supp(dρ) = E.

Definition. A set E is said to be a Denisov–Rakhmanov set if and only if

(i) E is essentially perfect and bounded. (ii) There is a set T_{E} compact in the uniform topology so that for any bounded

Jacobi matrix, J, for which (9.1) holds, the set of right limits of J lies in T_{E}.

The definition says nothing explicit about T_{E} being a torus, but by Proposition 9.1, if J_{r} ∈ T_{E}, then (9.2) holds, and since T_{E} is closed under translations, each J_{r} in T_{E} is almost periodic. By Kotani theory (see [60, 61, 95, 99]), ⟨δ_{n}, (J_{r} − x − iε)^{−1}δ_{n}⟩ has real boundary values for a.e x. In many cases and, in particular, if E is a finite union of closed intervals, Sodin–Yuditskii [104] (see also [5, 14]) proved there is a natural torus so that any almost periodic J_{r} with real boundary values lies in this torus. Thus for such cases, that E is a Denisov– Rakhmanov set can be connected to approach to an isospectral torus. In particular, our Theorem 1.2 implies the statement that σ_{ess}(J_{0}) is a Denisov–Rakhmanov set.

Given an essentially perfect set, E, we define D(E) to be the set of Jacobi matrices obeying (9.1).

The following two simple results will be the basis of our approximation theorems:

Proposition 9.2. Let E be an essentially perfect set. Suppose there are uniformly compact sets {T^{(n)}}_{n=1}^{∞} and T^{(∞)} of two-sided Jacobi matrices so that

(1) If J_{n} ∈ T^{(n)} and J_{n} → J_{∞} weakly, then J_{∞} ∈ T^{(∞)}.

(2) For any weak right limit point J_{r} of some J ∈ D(E), there is J̃ ∈ T^{(n)} so

sup_{|j|≤n} |a_{j}^{(r)} − ã_{j}| + |b_{j}^{(r)} − b̃_{j}| ≤ \frac{1}{n}  (9.3)

Then E is a Denisov–Rakhmanov set.

Proof. Let J_{n} be the J̃ guaranteed by (9.3). Then clearly, J_{n} converges weakly to J_{r} so, by (1), J_{r} ∈ T^{(∞)}. Since T^{(∞)} is uniformly compact, E is a Denisov– Rakhmanov set. □

Proposition 9.3. Let J_{0} be a fixed periodic Jacobi matrix with essential spectrum E_{0}. Then for any n, there is a δ > 0 so that for any set E with

(a)  E ⊂ {x : dist(x, E_{0}) < δ}  (9.4)
(b)  |E| > (1 − δ)|E_{0}|  (9.5)

and any J ∈ D(E), we have that any right limit, J_{r}, obeys (9.3) for some J̃ ∈ T_{J_{0}}.

Moreover, if p is fixed and C is a compact subset of [(0, ∞) × ℝ]^{p}, then δ can be picked to work for all J_{0} = {(a_{n}, b_{n})}_{n=1}^{p} ∈ C.

Proof. The uniformity claimed in the last statement comes from noting that choices can be made uniformly in the proof below.

Let p be the period of J_{0}. We first claim that given δ_{1}, we can find δ so if E obeys (9.4)–(9.5) then

dist(Δ(E), [−2, 2]) < δ_{1}  (9.6)
|{x ∈ (−2, 2) : all p solutions of Δ(y) = x lie in E}| > 4 − δ_{1}  (9.7)

This is immediate from the continuity of Δ and its derivatives.

Next, we note that given ε_{1}, we can find δ_{1} so that if J is a p × p block Jacobi matrix so that

σ_{ess}(J) ⊂ [−2 − δ_{1}, 2 + δ_{1}]
|{x ∈ [−2, 2] : J has a.c. spectrum at x of multiplicity p}| > 4 − δ_{1}

then

lim sup_{k,m→∞} |J_{km} − (S^{p} + S^{−p})_{km}| < ε_{1}

The proof of this is identical to the proof of the matrix Denisov–Rakhmanov theorem.

Combining these steps, we are reduced to showing for any n and ε, there is ε_{1} so for all two-sided J_{r} with dist(σ(J_{r}), E) < ε, we have that

sup_{k,m} |Δ(J_{r}) − (S^{p} + S^{−p})_{km}| < ε_{1}  (9.8)

implies there is a J̃ ∈ T_{J_{0}} so that (9.3) holds. To do this, we first follow the proof of Theorem 3.1 to note that for n, ε_{2}, and ε_{3} fixed, we can find ε_{1} so (9.8) implies there is a period p J^{♯}

∥Δ(J^{♯}) − Δ(J)∥ < ε_{2}  (9.9)

and

sup_{|j|≤n} |a_{j}^{(r)} − a_{j}^{♯}| + |b_{j}^{(r)} − b̃_{j}| ≤ ε_{3}

Finally, a compactness argument shows that for any n, we can find ε_{4} so for any periodic J^{♯} with

∥Δ(J^{♯}) − (S^{p} + S^{−p})∥ < ε_{4}

there is a J̃ ∈ T_{J_{0}} so that

∥J^{♯} − J̃∥ ≤ \frac{1}{2n}

Putting these together implies (9.3). □

Theorem 9.4. Let ℓ_{1}, ℓ_{2}, … be an arbitrary sequence in (2, 3, 4, …). For any ℓ_{1}- periodic Jacobi matrix J^{(0)}, there exist k_{2}, k_{3}, … so that for any limit periodic J with Jacobi coefficients

a_{n} = a_{n}^{(0)} + ∑_{m=2}^{∞} Re[A_{m}e^{2πin/ℓ_{1}ℓ_{2}...ℓ_{m}}]  (9.10)

b_{n} = b_{n}^{(0)} + ∑_{m=2}^{∞} Re[B_{m}e^{2πin/ℓ_{1}...ℓ_{m}}]  (9.11)

obeying

|A_{m}| + |B_{m}| ≤ k_{m}  (9.12)

we have that σ(J) is a Denisov–Rakhmanov set.

Remark. The study of limit periodic discrete Schrödinger operators with small tails was initiated by Avron–Simon [7] and Chulaevsky [19]. They prove purely a.c. spectrum.

Proof. As in [7, 19], one can pick the k_{m}’s so the spectrum is purely a.c. and so that the union of all isospectral tori for the periodic approximates lie in a fixed ℓ^{∞} compact sets. This implies the limit periodic potentials also have compact isospectral sets, and within this compact set, weak convergence implies norm convergence so hypothesis (1) of Proposition 9.2 holds. By decreasing the k_{m}’s if necessary, Proposition 9.3, continuity of the spectrum in ℓ^{∞} norm, and absolute continuity of periodic spectrum imply we can be sure that (9.3) holds. Thus Proposition 9.2 implies this theorem. □

Our final theorem in this section is the following:

Theorem 9.5. Fix ℓ. Let G = {(α_{1}, β_{1}, α_{2}, …, α_{ℓ+1}, β_{ℓ+1}) ∈ ℝ^{2ℓ+2} : α_{1} < β_{1} < α_{2} < β_{2} < ⋯ < β_{ℓ+1}}. For (α⃗, β⃗) ∈ G, define

E(α⃗, β⃗) = ⋃_{j=1}^{ℓ+1} [α_{j}, β_{j}]  (9.13)

Then {(α⃗, β⃗) : E(α⃗, β⃗) is a Denisov–Rahkmanov set} contains a dense G_{δ}.

Remarks. 1. As we have seen, the E(α⃗, β⃗) which arise from periodic problems is precisely those where the harmonic measure of each e_{j} = [α_{j}, β_{j}] is rational. In particular, if we fix α⃗ and β_{ℓ+1}, the set of β’s that are periodic is countable, and so certainly not a G_{δ}. We show that the family that leads to Denisov–Rahkmanov sets is uncountable.

2. It is a reasonable conjecture that every E is a Denisov–Rakhmanov set, so this result is weak. We include it because it is such a ‘cheap’ way to go beyond the periodic case using only that case.

Proof. For each (α⃗, β⃗), it is known [104] that there is an isospectral torus T of almost periodic J’s where (whole line) spectrum is precisely E(α⃗, β⃗). It follows from the construction in [104] that if (α^{(n)}, β⃗^{(n)}) ∈ G converge to (α^{(∞)}, β⃗^{(∞)}) ∈ G, then condition (1) of Proposition 9.2 holds.

Let G_{p} be the subset of G coming from periodic problems—this is dense in G. For (α^{(0)}, β⃗^{(0)}) ∈ G_{p}, pick J(α^{(0)}, β⃗^{(0)}) periodic with E(α^{(0)}, β⃗^{(0)}) as spectrum and pick δ_{n}(α^{(0)}, β⃗^{(0)}) via Proposition 9.3 requiring δ_{n} < \frac{1}{2} min(|β_{j}^{(0)} − α_{j}^{(0)}|, |α_{j+1}^{(0)} − β_{j}^{(0)}|). Let U^{(n)}(α⃗_{0}, β⃗_{0}) = {(α⃗, β⃗) : E(α⃗, β⃗) obeys (9.4)/(9.5) for E = E(α⃗, β⃗), E_{0} = E(α^{(0)}, β⃗^{(0)}) and δ = δ_{n}}, and let

U^{(n)} = ⋃_{G_{p}} U^{(n)}(α^{(0)}, β^{(0)})

This is dense and open. Then ∩_{n}U^{(n)} is a dense G_{δ} whose points, by construction and Proposition 9.2, correspond to Denisov–Rakhmanov sets. □

10. Sum Rules for MOPRL

In this section, our main goal is to prove the following two theorems about block Jacobi matrices:

Theorem 10.1 (P_{2} Sum Rule for MOPRL). Let J be a block Jacobi matrix with ℓ × ℓ Jacobi parameters {A_{n}}_{n=1}^{∞}, {B_{n}}_{n=1}^{∞} and matrix measure dρ. Let m(E) be given by (6.25) and suppose σ_{ess}(J) = [−2, 2]. Define for z ∈ 𝔻 \ {z = E + E^{−1} : E ∈ σ(J) \ [−2, 2]}

M(z) = −m(z + z^{−1})  (10.1)

Let F, G be the functions

F (β + β^{−1}) = \frac{1}{4} [β^{2} − β^{−2} − log β^{2}]  (10.2)

for β ∈ ℝ \ [−1, 1], that is, E = β + β^{−1} ∈ ℝ \ [−2, 2] and

G(a) = a^{2} − 1 − log(a^{2})  a ∈ (0, ∞)  (10.3)

Then lim_{r↑1} M(re^{iθ}) exists for a.e. θ and

\frac{1}{2π} ∫ log(\frac{sin^{ℓ} θ}{det(Im M(e^{iθ}))}) sin^{2} θ dθ

+ ∑_{E∈σ(J)\[−2,2]} F (E) = ∑_{n=1}^{∞} Tr(\frac{1}{4} B_{n}^{2} + \frac{1}{2} G(|A_{n}|))

(10.4)

Remarks. 1. All terms are positive (since F and G are positive, this is evident for two terms; positivity of the integral will be seen below), so this sum rule always makes sense, although some terms may be +∞.

2. Recall that |A_{n}| = \sqrt{A_{n}^{∗}A_{n}}; although since the formula for G(a) only involves a^{2}, one does not need to take a square-root.

3. Because of the trace and absolute value, Tr(\frac{1}{4}B_{n}^{2} + \frac{1}{2}G(|A_{n}|)) is constant over equivalence classes of Jacobi matrix parameters.

4. In the type 1 case, the RHS of (10.4) is finite if and only if J − S^{p} − S^{−p} is Hilbert–Schmidt. This is also true when J is of type 3; see Proposition 11.11.

Theorem 10.2 (Sharp Case C_{0} Sum Rule for MOPRL). Consider the three quantities:

Z(J) = \frac{1}{4π} ∫_{0}^{2π} log(\frac{sin^{ℓ} θ}{det(Im M(e^{iθ}))}) dθ  (10.5)
E_{0}(J) = ∑_{E∉σ(J)\[−2,2]} log(|β|)  (10.6)

where β is related to E by

β ∈ ℝ \ [−1, 1]  E = β + β^{−1}  (10.7)

and

A_{0}(J) = lim_{N→∞} − ∑_{n=1}^{N} log(det(|A_{n}|))  (10.8)

which we suppose exists but it may be +∞ or −∞. Then

(i) If any two of Z, E_{0}, A_{0} are finite, then so is the third. (ii) If all are finite, then

Z(J) = A_{0}(J) + E_{0}(J)  (10.9)

(iii) If all are finite, then

lim_{N→∞} ∑_{n=1}^{N} Tr(B_{n})  (10.10)

exists.

Remark. We will prove (and actually use it to prove Theorem 1.3) that if E_{0}(J) < ∞, then Z(J) < ∞ so long as

\underline{A}_{0}(J) = lim inf_{N}(− ∑_{n=1}^{N} log(det(|A_{n}|))) < ∞  (10.11)

Theorem 10.1 is a matrix-valued analog of the OPRL P_{2} sum rule of Killip–Simon [59], and Theorem 10.2 of the OPRL Case C_{0} sum rule by Simon–Zlatoš [103]. Both were refinements of sum rules of Case [17, 18] who in turn was motivated by earlier KdV and Toda sum rules. Case only considered short-range |a_{n} − 1| + |b_{n}|, while [59, 103] considered the necessary techniques to go up to the borderline of validity.

[103] had some simplifications of [59], and [96] further simplified, although each of the later two proofs depends heavily on the earlier ones. Here, following Simon [96], we will prove a nonlocal step-by-step sum rule. As there, the key is a suitable representation theorem for meromorphic Herglotz functions—in this case, extended to matrix-valued functions.

For a ∈ (−1, 1), we define Blaschke factors as usual by

b(z, a) = \begin{cases}\frac{a−z}{1−az} & 0 < a < 1 \\ \frac{z−a}{1−az} & −1 < a ≤ 0 \end{cases}  (10.12)

Proposition 10.3. Let f(z) be an ℓ × ℓ matrix-valued meromorphic function on 𝔻 so that (Im A ≡ (A − A^{∗})/2i)

(i)  ± Im f(z) > 0 when ± Im z > 0  (10.13)
(ii)  lim_{z→0} f(z)z^{−1} = 1  (10.14)

Then

(a) For a.e. θ, lim_{r↑1} f(re^{iθ}) ≡ f(e^{iθ}) exists.

(b) log|det(f(e^{iθ}))| ∈ ∩_{1≤p<∞}L^{p}(∂𝔻, dθ/2π)

(c) All the zeros and poles of det(f(z)) lie on (−1, 1) and are of finite order. Let {z_{j}}_{j=1}^{∞} and {p_{j}}_{j=1}^{∞} be those zeros and poles of det(f(z)) repeated up to multiplicity (it can also happen that both sets are finite). z = 0 is not included in {z_{j}}. Then

B_{∞}(z) = lim_{r↑1} \frac{∏_{|z_{j}|<r} b(z, z_{j})}{∏_{|p_{j}|<r} b(z, p_{j})}  (10.15)

exists and obeys:

(i) B_{∞} is analytic and nonvanishing on ℂ\{z_{j}}∪{p_{j}}∪{z_{j}^{−1}}∪{p_{j}^{−1}}∪{±1} (ii) |B_{∞}(e^{iθ})| = 1 on ∂𝔻 \ {±1} (iii)

|arg(B_{∞}(z))| ≤ 2πℓ  (10.16)

for |z| < 1 with arg normalized by arg B_{∞}(0) = 0.

(d) We have the representation

det(f(z)) = z^{ℓ}B_{∞}(z) exp(∫ \frac{e^{iθ} + z}{e^{iθ} − z} log|det(f(e^{iθ}))| \frac{dθ}{π} )  (10.17)

Remarks. 1. It should be possible to prove that 0 < arg(B_{∞}(z)) < πℓ for Im z > 0; we settle for the weaker result.

2. (10.14) is not central for a result of this type, but it is true in applications and simplifies the notation.

3. This result for ℓ = 1 is in [96]. ℓ > 1 has some subtleties, but the basic strategy we use is that of [96].

We will prove this result in a sequence of lemmas:

Lemma 10.4. det(f(z)) is analytic and nonvanishing in Ω ≡ {z : z ∈ 𝔻, Im z > 0}, and arg(det(f(z))) can be chosen in that region to be continuous so that

0 < arg(det(f(z))) < πℓ  (10.18)

Proof. In Ω, all matrix elements ⟨φ, f(z)φ⟩ are analytic and have a.e. boundary values (since they are scalar Herglotz functions), so by polarization, f(z) is analytic on Ω and has a.e. boundary values. Thus det(f(z)) as a polynomial in matrix elements is analytic on Ω.

Consider

P (λ, z) = det(λ1 − f(z))  (10.19)

which is a polynomial in λ with analytic coefficients away from the poles of f. It follows, for z near any z_{0} about which f is analytic, that the roots P (λ, z) = 0 written as a function of z are analytic functions in (z − z_{0})^{1/k} for some k depending on z_{0}. It then follows that near any fixed z_{0}, all roots are analytic, that is, singularities are isolated.

Pick x_{0} ∈ (0, ε) so that x_{0}f(x_{0}) > 0, so all eigenvalues λ_{1}(x_{0}), …, λ_{ℓ}(x_{0}) are in (0, ∞). Let z ∈ Ω be a point about which all eigenvalues are analytic, and let γ(z) be a simple closed path from x_{0} to z which avoids the discrete set where eigenvalues are not analytic and lie in Ω except for x_{0} with, say, γ(0) = x_{0}, γ(1) = z. By analytically continuing eigenvalues, we get function {λ_{j}(z)}_{j=1}^{ℓ}, so λ_{j}(z) are all the eigenvalues of f(γ(t)) and λ_{j}(0) ∈ (0, ∞). By Im f > 0, Im λ_{j}(z) > 0, so if we define arg(λ_{j}(z)) with arg(λ_{j}(0)) = 0, we have

0 < arg(λ_{j}(z)) < π

Thus

arg(det(f(z))) = ∑_{j=1}^{ℓ} arg(λ_{j}(z))

normalized by arg(det(f(x_{0}))) = 0 obeys (10.18).

By analyticity of det(f(z)) and the fact that it is nonvanishing, arg(det(f(z))) is uniquely defined as a continuous function on Ω with lim_{ε↓0} arg(det(f(x_{0} +iε))) = 0. By the above, (10.18) holds at all points z in Ω where all eigenvalues are analytic and so, by continuity and the open mapping theorem for analytic functions, all points. □

Lemma 10.5. Let a < b lie in (−1, 1) so that both a and b are neither a zero nor a pole of det(f(z)). Let Z(a, b), P (a, b) be the number of zeros, poles of det(f(z)) in (a, b) counting multiplicity. Then

|Z(a, b) − P (a, b)| ≤ ℓ  (10.20)

Proof. By the argument principle, 2π(Z − P) is the change of arg(det(f(z)) along the circle through a and b centered at \frac{1}{2}(a + b). By Lemma 10.4, this is at most 2(ℓπ). □

Lemma 10.6. The sets of zeros and poles (with multiplicity) of det(f(z)), including the ℓ-fold zero at z = 0, can be written as ℓ subsets z_{j}^{(k)}, p_{j}^{(k)} with k = 1, …, ℓ and −Ñ_{k} < j < N_{k} (with N_{k} and Ñ_{k} among 1, 2, …, ∞) so that z_{0}^{(k)} = 0 and

z_{j}^{(k)} < p_{j}^{(k)} < z_{j+1}^{(k)}  (10.21)

for all allowed values of j.

Remarks. 1. If there are infinitely many z in (−1, 0) and in (0, 1), then Ñ_{k} = N_{k} = ∞ for all k. The awkwardness requiring N_{k}, Ñ_{k} is only needed if there are finitely many zeros.

2. To avoid notational complexity, we slightly lied if N_{k} or Ñ_{k} is finite. If N_{k} is finite, z_{j}^{(k)} runs to j = N_{k}. p_{j}^{(k)} can then run to either N_{k} or N_{k} − 1.

Proof. Construct S_{1}, S_{2}, …, S_{ℓ} as follows: Set z_{0}^{(1)} = 0. Let p_{0}^{(1)} be the first pole larger than z_{0}^{(1)}, z_{1}^{(1)} the first zero larger than p_{0}^{(1)}, p_{1}^{(1)} the next pole, etc. This either continues indefinitely, in which case we set N_{1} = ∞, or stops because there is no next zero or pole. Then do the same to the left of 0, that is, p_{−1}^{(1)} is the first pole smaller than z_{1}^{(1)}, etc. Clearly, the points in S_{1} obey (10.21). Now remove the points of S_{1} (or decrease their multiplicity by 1) and repeat the construction (starting with z_{0}^{(2)} = 0) to make S_{2}, S_{3}, …, S_{ℓ}.

We claim that after we construct ℓ S_{j}’s, we have exhausted all the poles and zeros. Let us show this is true for (0, 1); the argument for (−1, 0) is similar (and since 0 has multiplicity ℓ, it is removed after ℓ steps).

Suppose z̃ is a zero that is left and it is closer to zero than any leftover zero or pole. If z̃ lies in some (p_{j}^{(k)}, z_{j+1}^{(k)}), j = 0, 1, …, we could have used it as z_{j+1}^{(k)} so it cannot lie in any such interval. Put differently, there are only matched zeros and poles in (0, z̃) ∩ ∪_{j=1}^{ℓ}S_{j}. By the choice of z̃, there are no other poles in (0, z̃). Thus, for small δ, the interval (−δ, z̃+ δ) has ℓ + 1 extra zeros over poles, violating Lemma 10.5. So the closest leftover point is not a zero.

Suppose p̃ is a pole that is left and it is closer to zero than any other leftover zero or pole. As above, p̃ cannot lie in any (z_{j}^{(k)}, p_{j}^{(k)}), j = 0, 1, …, so there are only matched zeros and poles in [0, p̃) ∩ ∪_{j=1}^{ℓ}S_{j}. But then, for small δ, (δ, p̃ + δ) has ℓ + 1 extra poles, violating Lemma 10.5. Thus ∪_{j=1}^{ℓ}S_{j} includes all zeros and poles. □

Lemma 10.7. The limit B_{∞}(z) of (10.15) exists and obeys conditions (i)–(iii) of Proposition 10.3(c).

Proof. Renumber the p_{j}^{(k)} into a single sequence p_{1}, p_{2}, …, so |p_{1}| ≤ |p_{2}| ≤ ⋯ and let z_{m} be the correspondingly paired z_{j+1}^{(k)} (paired to the p_{j}^{(k)} that is p). Since {(p_{j}^{(k)}, z_{j+1}^{(k)})}_{j=1}^{N_{j}} are disjoint subsets of (0, 1) for each fixed k,

∑_{j=1}^{∞} |z_{j+1}^{(k)} − p_{j}^{(k)}| = ∑_{j=1}^{∞} z_{j+1}^{(k)} − p_{j}^{(k)} < 1

so we see that

∑_{j=1}^{∞}|z_{j} − p_{j}| ≤ 2ℓ

The existence of B_{∞} then follows by Proposition 13.8.2 of [99], as do (i) and (ii).

To get (iii), we note that just taking the zeros and poles in a single S_{j} yields a set obeying (13.8.5) and (13.8.6) of [99]. So, by (13.8.10), that product has arg bounded by 2π. The ℓ-fold product thus obeys (10.16). □

Proof of Proposition 10.3. Given Lemma 10.7, the proof is essentially that of Theorem 13.8.3 of [99].

g(z) ≡ \frac{det(f(z))}{z^{ℓ}B_{∞}(z)}  (10.22)

is analytic and nonvanishing on 𝔻 with g(0) > 0 (since B_{∞}(0) > 0). Moreover, by (10.18) and (10.16),

|arg g(z)| ≤ 4πℓ  (10.23)

so, by M. Riesz’ theorem, log(g(z)) ∈ ∩_{p<∞}H^{p}(∂𝔻) from which (a), (b) of the theorem are immediate and (d) follows from the Poisson representation for log(g(z)) since log(|g(e^{iθ})|) = log(|det(f(e^{iθ}))|. □

Now we turn to block Jacobi matrices where we obtain:

Theorem 10.8 (Nonlocal Step-by-Step Sum Rule for Block Jacobi Matrices). Let J be a block Jacobi matrix with σ_{ess}(J) ⊂ [−2, 2] and Jacobi parameters {A_{n}, B_{n}}_{n=1}^{∞}. Let J^{(1)} denote this Jacobi matrix with the top row of blocks and left-most column of blocks removed. Let m(E), m^{(1)}(E) be the m-functions given by (6.25). Let M, M^{(1)} be defined on 𝔻 by

M(z) = −m(z + z^{−1})  (10.24)

with poles at {p_{i}}_{i=1}^{N} where p_{i} + p_{i}^{−1} are eigenvalues of J. We repeat each p_{i} a number of times equal to the multiplicity of the eigenvalues (equivalently, the rank of the residue). Let {z_{i}}_{i=1}^{N−1} be the corresponding points for J^{(1)}. Then

(a) The Blaschke product, B_{∞}(z), defined by the {z_{i}} ∪ {p_{i}} via (10.15) exists and obeys (i)–(iii) of Proposition 10.3(c).

(b) M(z) and M^{(1)}(z) have limits M(e^{iθ}) and M^{(1)}(e^{iθ}) as r ↑ 1 for z = re^{iθ} for a.e. θ in ∂𝔻 and

log[\frac{det(Im M)(e^{iθ})}{det(Im M^{(1)}(e^{iθ}))}] ∈ ⋂_{1≤p<∞} L^{p}(∂𝔻, \frac{dθ}{2π} )  (10.25)

(c)

det(\frac{|A_{1}|M(z)}{z}) = B_{∞}(z) exp(∫ \frac{e^{iθ} + z}{e^{iθ} − z} log[\frac{det(Im M(e^{iθ}))}{det(Im M^{(1)}(e^{iθ}))}]\frac{dθ}{4π} ) (10.26)

Remark. As in the case ℓ = 1, it can happen (although not in examples where sum rules are finite) that det(Im M(e^{iθ})) = det(Im M^{(1)}(e^{iθ})) = 0 for θ in a set of positive measure. (b) and (c) are shorthand for the more precise

(i) For a.e. θ, det(Im M(e^{iθ})) = 0 if and only if det(Im M^{(1)}(e^{iθ})) = 0. (ii) There is an a.e. positive function g(θ) on ∂𝔻, equal to det(Im M(e^{iθ}))/ det(Im M^{(1)}(e^{iθ}))

when the ratio is not 0/0 so that (10.25) and (10.26) hold if the formal ratio is replaced by g(θ).

Proof. Given Proposition 10.3, this is essentially identical to the proof of Theorem 13.8.4 of [99] with care given to matrix issues. We begin by noting that (6.31) for n → n + 1 first implies near z = 0

M^{(n+1)}(z)^{−1} = z^{−1} + O(1)  (10.27)

and then by (6.25) that

(\frac{M^{(n)}(z)}{z})^{−1} = 1 − B_{n+1}z − (A_{n+1}^{∗}A_{n+1} − 1)z^{2} + O(z^{3})  (10.28)

Since M^{(n)}(z)/z is near 1 for z small, we can compute its determinant using

det(C) = exp(Tr(log(C))  (10.29)

which holds if ∥C − 1∥ < 1. Thus

log det(\frac{M^{(n)}(z)}{z})
= Tr(B_{n+1})z + [Tr\{([A_{n+1}^{∗}A_{n+1} − 1] + \frac{1}{2}B_{n+1}^{2})\}]z^{2} + O(z^{3})

(10.30)

In addition, (6.31) implies

Im[M(z)^{−1}] = Im(z + z^{−1}) − A_{1} Im M_{1}(z)A_{1}^{∗}  (10.31)

so at points where M(z) has radial limits (a.e. θ, see below),

−[M(e^{iθ})^{∗}]^{−1} Im M(e^{iθ})[M(e^{iθ})]^{−1} = −A_{1} Im M(e^{iθ})A_{1}^{∗}  (10.32)

which, using (on account of det(|C|)^{2} = det(C^{∗}) det(C))

|det(A_{1})| = det(|A_{1}|)

yields

|det(|A_{1}|M(e^{iθ}))|^{2} = \frac{det(Im M(e^{iθ}))}{det(Im M_{1}(e^{iθ}))}  (10.33)

We now apply Proposition 10.3 to M(z) which obeys (10.13) (since Im(z+z^{−1}) < 0 on 𝔻 and (10.24) has a minus sign) and (10.14) by (10.27).

By Theorem 6.7, our B_{∞}(z) here (after perhaps canceling some zeros and poles) is the B_{∞}(z) of Proposition 10.3. (a) and (b) immediately follow from Proposition 10.3. We get (10.26) from (10.17) by using (10.33) (noting (10.26) has a 1/4π while (10.17) a 1/2π on account of the square on the left side of (10.33)). We also use that if c is a positive constant,

exp(∫ \frac{e^{iθ} + z}{e^{iθ} − z} log(c^{2})\frac{dθ}{4π} ) = c  (10.34)
□

As in [96], we can get step-by-step P_{2} (originally in [59]), C_{0}, C_{1} (originally in [103]) sum rules immediately from Taylor expansion of the log of (10.26). We let β_{j}(J) be the numbers in (−1, 1) \ {0} for which E_{j} ≡ β_{j} + β_{j}^{−1} are eigenvalues of J counting multiplicities.

Theorem 10.9 (C_{0}, C_{1}, P_{2} Step-by-Step Sum Rules). (i)

\frac{1}{4π} ∫_{0}^{2π} log(\frac{det(Im M^{(1)}(e^{iθ}))}{det(Im M(e^{iθ}))})dθ
− ∑_{j} log(|β_{j}(J)|) − log(|β_{j}(J^{(1)})|) = − log(det|A_{1}|)

(10.35)

(ii)

−\frac{1}{2π} ∫_{0}^{2π} log(\frac{det(Im M^{(1)}(e^{iθ}))}{det(Im M(e^{iθ}))}) cos θ

+ ∑_{j}[β_{j}(J) − (β_{j}^{−1}(J)^{−1})] − [β_{j}(J^{(1)}) − (β_{j}(J^{(1)}))^{−1}] ≡ Tr(B_{1})

(10.36)

(iii)

\frac{1}{4π} ∫ log(\frac{det(Im M^{(1)}(e^{iθ}))}{det(Im M(e^{iθ}))}) sin^{2}(θ)ϕ(θ)
+ ∑ F (E_{j}(J)) − F (E_{j}(J^{(1)})) = Tr\frac{1}{4}(B_{1}^{2} + \frac{1}{2} G(|A_{1}|))

(10.37)

where F is given by (10.2) and G by (10.3).

Remark. The E_{j}(J) in (−∞, −2) and (2, ∞) and E_{j}(J^{(1)}) interlace in the ℓ = 1 case. In the general ℓ case, we have at most ℓ fewer eigenvalues of J^{(1)} on any (−∞, −E_{0}) or (E_{0}, ∞) so, as in Lemma 10.6, one can decompose into ℓ interlacing subsets. This and the monotonicity of functions like F show the eigenvalue sums in (10.35)–(10.37) are conditionally convergent. Similarly, the integrals are always convergent.

Proof. Apply log to both sides of (10.26) and take Taylor coefficients. The constant term is (10.35) and the first derivative is (10.36). If L(z) is the log of the left side and R(z) of the right, then

L(0) + \frac{1}{2} L′′(0) = R(0) + \frac{1}{2} R′′(0)

is (10.37). □

The proofs of Theorems 10.1–10.2 are now identical to those of the scalar case; see, for example, the discussion of Theorems 13.8.6 and 13.8.8 of [99]. In particular, Z(J) and Q(J) (the integral on the right of (10.4)) are negatives of relative entropies, and so, lower semi-continuous.

11. Szegő and Killip–Simon Theorems When All Gaps Are Open

Our goal here is to prove Theorems 1.3 and 1.4. Our strategy, of course, will be to translate Theorems 10.1 and 10.2 for Δ(J) to statements about J. Firstly, we need to relate the a.c. part of the matrix measure for Δ(J) to the a.c. part of the (scalar) measure for J. And secondly, to relate ℓ^{2} norms of coefficients of Δ(J) to the distance of J’s Jacobi parameters to the isospectral torus. We begin with the first question. Thus we take

dρ_{J}(x) = ω(x) dx + dμ_{J,s}(x)  (11.1)

with dμ_{J,s} singular and ω supported precisely on σ_{ess}(J_{0}). By this assumption and the spectral mapping theorem, Δ(J) has a.c. spectrum precisely on [−2, 2] so the matrix measure for Δ(J) has the form

dρ_{Δ(J)}(E) = W (E) dE + dρ_{Δ(J),s}(E)  (11.2)

Proposition 11.1. Let J_{0} be a periodic Jacobi matrix with period p and J a Jacobi matrix with Jacobi parameters {a_{n}, b_{n}}_{n=1}^{∞} and measure dρ_{J} of the form (11.1) with ω supported on σ_{ess}(J_{0}). Let Δ be the discriminant for J_{0} and W (E) the a.c. part of the p × p matrix-valued measure dρ_{Δ(J)} associated to Δ(J) (so W is a p × p matrix). Then for E ∈ (−2, 2) and Δ^{−1}(E) = {x_{1}, …, x_{p}},

det(W (E)) = ( ∏_{j=1}^{p} a_{j}^{2p−2j}[a_{j}^{(0)}]^{p})(∏_{j=1}^{p} ω(x_{j}))  (11.3)

Proof. In the block Jacobi form, dρ_{Δ(J)} has jk matrix element equal to the spectral measure of the operator Δ(J) associated to δ_{j}, δ_{k}, that is, ∫ F (x)(dρ_{Δ}(x))_{jk} = ⟨δ_{j}, F (Δ(E))δ_{k}⟩. But δ_{j} = p_{j−1}(J)δ_{1}. It follows that

W_{kj}(E) = ∑_{ℓ=1}^{p} ω(x_{ℓ})(|Δ′(x_{ℓ})|)^{−1}p_{k−1}(x_{ℓ})p_{j−1}(x_{ℓ})  (11.4)

Note that the factors of 1/Δ′ arise from the Jacobian \frac{dE}{dx} = Δ′(x). We can re-write (11.4) as W_{kj}(E) = (MAM^{t})_{kj} where A is the diagonal matrix

A_{ℓm} = δ_{ℓm}ω(x_{ℓ})(|Δ′(x_{ℓ})|)^{−1}  (11.5)

and M is the matrix

M_{kℓ} = p_{k−1}(x_{ℓ})  k = 1, …, p; ℓ = 1, …, p  (11.6)

Next we compute det(M); det(A) is easy. Note that

p_{k−1}(x_{ℓ}) = ( ∏_{j=1}^{k−1} a_{j})^{−1}x_{ℓ}^{k−1} + lower order  (11.7)

Moreover, inductively one sees that the lower order terms can be neglected in the determinant—they can be removed by subtracting a multiple of rows above (i.e., smaller values of k). Thus,

det(M) = [ ∏_{k=1}^{p}( ∏_{j=1}^{k−1} a_{j})^{−1}] det(x_{ℓ}^{k−1}) = (∏_{j=1}^{p} a_{j}^{p−j})(∏_{j>k}(x_{j} − x_{k})) (11.8)

by the well-known formula for Vandermonde determinants. This can be simplified further. The points x_{j} are precisely the zeros of the polynomial Δ(x) − E; hence, invoking (2.14),

Δ(x) − E = (∏_{j=1}^{p} a_{j}^{(0)})^{−1}(∏_{k=1}^{p}(x − x_{k}))

In this way we discover that

det(M)^{2} = (∏_{j=1}^{p} a_{j}^{2p−2j} [a_{j}^{(0)}]^{p})(∏_{k=1}^{p} |Δ′(x_{k})|)  (11.9)

Multiplying this by det(A) gives (11.3). □

Corollary 11.2. If J_{0} has all gaps open and α > −1, then

∫_{−2}^{2}(4 − E^{2})^{α}|log det(W (E))| dE < ∞  (11.10)

if and only if

∫_{σ_{ess}(J_{0})} dist(x, ℝ \ σ_{ess}(J_{0}))^{α}|log ω(x)| dx < ∞  (11.11)

When α = −\frac{1}{2}, the same conclusion holds even if some gaps are closed.

Remark. Since α > −1, (4 − E^{2})^{α} (resp., dist(…)^{α}) are in L^{p} for some p > 1, so the log_{+}( ) is always integrable and these conditions are equivalent to the integral without |·| being larger than −∞.

Proof. Changing variables via E = Δ(x) and applying Proposition 11.1 shows that (11.10) holds if and only if

∫_{σ_{ess}(J_{0})} | log ω(x)| (4 − Δ(x)^{2})^{α} |Δ′(x)| dx < ∞  (11.12)

If all gaps are open then |Δ′(x)| is strictly positive on σ_{ess}(J_{0}), while 4 − Δ(x)^{2} is a polynomial with a simple zero at each band edge (and no others). This proves the first claim.

At a closed gap, 4 − Δ(x)^{2} has a double zero and Δ′(x) a simple zero. When α = −\frac{1}{2}, these cancel exactly. □

Next we turn to the ℓ^{2} issue. Given any two-sided periodic matrix J̃ with Jacobi parameters {a_{n}, b_{n}}_{n=1}^{p} and fixed periodic J_{0}, let B_{J_{0}}(J̃), A_{J_{0}}(J̃) be the constant p×p blocks in Δ_{J_{0}}(J̃). We are heading towards showing that ∥B_{J_{0}}(J̃)∥_{2}^{2}+∥A_{J_{0}}(J̃)− 1∥_{2}^{2} is comparable to dist((a_{n}, b_{n})_{n=1}^{p}, T_{J_{0}}). This will be the key to showing ℓ^{2} tails in the matrix pieces of Δ(J) − S^{p} − S^{−p} for general J is equivalent to (1.28).

The crucial fact will be that the polynomial coefficients of Δ_{J_{0}} − ΔJ̃ are comparable to dist((a_{n}, b_{n})_{n=1}^{p}, T_{J_{0}}). For this we need the following, which is a simple application of the implicit function theorem and compactness:

Lemma 11.3. Let F be a C^{∞} map of an open set U ⊂ ℝ^{n} to ℝ^{ℓ} with ℓ < n. Suppose T = F^{−1}(y_{0}) is a smooth manifold of dimension n − ℓ and compact for some y_{0} ∈ ℝ^{ℓ}, and

rank((∇F)(x_{0})) = ℓ  (11.13)

for all x_{0} ∈ T . Then for any compact neighborhood, K, of T , there are c_{k}, d_{k} ∈ (0, ∞) so for all x ∈ K,

c_{k}|F (x) − y_{0}| ≤ dist(x, T ) ≤ d_{k}|F (x) − y_{0}|  (11.14)

One can restate (11.13) in a more illuminated way in terms of the components F_{1}, …, F_{ℓ} of T . Of course, ∇F_{j}(x_{0}) is orthogonal to T at x_{0}. The condition (11.13) is equivalent to saying that {∇_{j}F (x_{0})}_{j=1}^{ℓ} span the normal bundle to T . This is equivalent to saying they are linearly independent. Notice that if J_{0} has all gaps open, T_{J_{0}} is of dimension p − 1 = 2p − (p + 1) and Δ_{J_{0}} is a polynomial of degree p, hence with p + 1 coefficients. Thus the following shows we can use Lemma 11.3:

Theorem 11.4. Suppose all gaps are open for some periodic J_{0}. Then at any point in T_{J_{0}}, the gradients of the derivatives of the coefficients of Δ_{J} span the normal bundle of T_{J_{0}} in ℝ^{2p}.

Proof. Δ_{J_{0}} has the form

Δ_{J_{0}}(x) = (a_{1} … a_{p})^{−1} ∏_{j=1}^{p}(x − λ_{j}) = ∑_{j=0}^{p} c_{j}x^{j}

where λ_{j} are the roots. The coefficients thus obey

c_{p}^{−1} = a_{1} … a_{p}  (11.15)
c_{ℓ}c_{p}^{−1} = ∑_{1≤k_{1}≤···≤k_{p−ℓ}≤p} λ_{k_{1}} … λ_{k_{1−ℓ}} ≡ s_{k−ℓ}  ℓ < p  (11.16)

It is well known that if

t_{ℓ} = ∑_{j=1}^{p} λ_{j}^{ℓ}  (11.17)

then t_{ℓ} is ℓs_{ℓ} plus a polynomial in {s_{j}}_{j=1}^{ℓ−1}, so {∇t_{j}}_{j=1}^{ℓ} and {∇s_{j}}_{j=1}^{ℓ} span the same space. It follows that we need only show the gradients of c_{p}^{−1} and t_{ℓ} span the normal bundle of T_{J_{0}}.

Let

M_{0} = {(a_{n}, b_{n})_{ℓ=1}^{p} : a_{1}a_{2} … a_{p} = a_{1}^{(0)}a_{2}^{(0)} … a_{p}^{(0)}; b_{1} + ⋯ + b_{p} = b_{1}^{(0)} + ⋯ + b_{p}^{(0)}} (11.18)

We know M_{0} ⊃ T_{J_{0}}. Clearly, ∇c_{p}^{−1} and ∇t_{1} span the normal bundle to M_{0} since t_{1} = ∑_{j=1}^{p} b_{j} (see (2.14)). Thus we need only show the projections of {∇t_{ℓ}}_{ℓ=2}^{p} into the tangent space of M_{0} span the normal bundle of T_{J_{0}} in M_{0}.

Studies of the Toda flows show that M_{0} is a symplectic manifold with {t_{ℓ}}_{ℓ=2}^{p} Poisson commuting. Since the symplectic form on M_{0} is nondegenerate, to say {∇t_{j}}_{j=2}^{p} span the normal bundle is the same as saying that the Hamiltonian flows generated by {t_{j}}_{j=2}^{p} span the tangent bundle of T_{J_{0}}, or equivalently, given dim(T_{J_{0}}) = p − 1, that these Hamiltonian flows are independent.

This independence is a theorem of van Moerbeke [108, Theorem 5.2] or [102]. □

Lemma 11.5. For any compact subset, K, of period p Jacobi matrices, there exist constants c_{K} and d_{K} in (0, ∞) so for all J ∈ K,

c_{K} ∥ ∑_{ℓ=0}^{p} α_{ℓ}J^{ℓ}χ_{p}∥_{2} ≤ ∑_{ℓ=0}^{p} |α_{ℓ}|^{2} ≤ d_{K} ∥ ∑_{ℓ=0}^{p} α_{ℓ}J^{ℓ}χ_{p}∥_{2}  (11.19)

where χ_{p} is the projection onto the p-dimensional space spanned by {δ_{j}}_{j=1}^{p}.

Proof. {J^{ℓ}χ_{p}}_{ℓ=0}^{p} are independent since J^{ℓ} has strictly positive elements in the ℓ-th diagonal and {J^{k}}_{k<ℓ} only has zero elements there. Hence, the matrix

Tr(χ_{j}J^{ℓ}J^{k}χ_{p})|_{ℓ,k=1,...,p}  (11.20)

is strictly positive so (11.19) holds for each fixed J. The optimal constants are clearly continuous so uniformly bounded above and below on K. □

Proposition 11.6. Let J_{0} be a periodic Jacobi matrix with all gaps open. For any compact neighborhood K of T_{J_{0}} in (0, ∞)^{p} × ℝ^{p}, there are constants c_{K} and d_{K} in (0, ∞) so that for all J ∈ K,

c_{K}(∥A_{J_{0}}(J) − 1∥^{2} + ∥B_{J_{0}}(J)∥^{2})^{1/2} ≤ dist(J, T_{J_{0}})
≤ d_{K}(∥A_{J_{0}}(J) − 1∥^{2} + ∥B_{J_{0}}(J)∥^{2})^{1/2}

Proof. We have that

2∥A_{J_{0}}(J) − 1∥^{2} + ∥B_{J_{0}}(1)∥^{2} = ∥[Δ_{J_{0}}(J) − (S^{p} + S^{−p})]χ_{p}∥^{2}  (11.21)

But by the magic formula,

Δ_{J}(J) = S^{p} + S^{−p}  (11.22)

so

[Δ_{J_{0}}(J) − (S^{p} + S^{−p})]χ_{p} = ∑_{ℓ=0}^{p} c_{ℓ}J^{ℓ}χ_{p}  (11.23)

where c_{ℓ} is the difference of coefficients for J and J_{0}. By Lemma 11.5,

RHS of (11.21) ∼ ∑_{ℓ=0}^{p} |c_{ℓ}|^{2}  (11.24)

where ∼ means the ratio is bounded above and away from zero on compact subsets. By Lemma 11.3 and Theorem 11.4,

∑_{ℓ=0}^{p} |c_{ℓ}|^{2} ∼ dist(J, T_{J_{0}})^{2}  (11.25)

Combining this with (11.24) proves the proposition. □

Now we take a general J not periodic and form Δ_{J_{0}}(J) which is a one-sided block Jacobi matrix with block elements A_{n,J_{0}}(J), B_{n,J_{0}}(J).

Lemma 11.7. Δ_{J_{0}}(J)_{kℓ} for k ≤ ℓ depends only on {b_{j}}_{j=k−α}^{ℓ+α} and {a_{j}}_{j=k−α}^{ℓ+α−1} where α = ⌊\frac{1}{2}(p−(ℓ−k))⌋ is the greatest integer less than or equal to \frac{1}{2}[p−(ℓ−k)].

Proof. Each factor of J changes index by at most one. In order to get from k to ℓ, ℓ − k steps are needed. The remainder cannot go below ℓ − α or above k + α and get back to k in p steps. □

Lemma 11.8. Let J have Jacobi parameters {a_{n}, b_{n}}_{n=1}^{∞}. Let J̃ be periodic with period p and suppose b_{n} = b̃_{n} for kp − p ≤ n ≤ kp + 2p and a_{n} = ã_{n} for kp − p ≤ n ≤ kp + 2p − 1. Then

A_{k,J_{0}}(J) = A_{J_{0}}(J̃) B_{k,J_{0}}(J) = B_{J_{0}}(J̃)

Proof. Immediate from Lemma 11.7. □

Lemma 11.9. Let k ≤ ℓ and α = [\frac{1}{2}(p − (ℓ − k))]. For any two J and J̃ and any K, there is C_{K} so that

|Δ_{J_{0}}(J)_{kℓ} − Δ_{J_{0}}(J̃)_{kℓ}| ≤ C_{K} sup_{k−α≤j≤ℓ+α}[|b_{j} − b̃_{j}| + |a_{j} − ã_{j}|]  (11.26)

so long as

sup[|b_{j}| + |b̃_{j}| + |a_{j}| + |ã_{j}|] ≤ K  (11.27)

Proof. Immediate from Lemma 11.7 and the fact that Δ_{J_{0}} has matrix elements that are fixed (given J_{0}) polynomials in a’s and b’s. □

Lemma 11.10. If

∑_{k=1}^{∞} |A_{k,J_{0}}(J) − 1|^{2} + |B_{k,J_{0}}(J)|^{2} < ∞  (11.28)

then

∑_{k=1}^{∞} |A_{k+1,J_{0}}(J) − A_{k,J_{0}}(J)|^{2} + |B_{k+1,J_{0}}(J) − B_{k,J_{0}}(J)|^{2} < ∞  (11.29)

Proof. The difference of two ℓ^{2} vectors is in ℓ^{2}. □

Our next preliminary is to relate A ∈ L to

|A| = \sqrt{A^{∗}A}  (11.30)

Proposition 11.11. The map A ↦ |A| from L to positive definite matrices is a diffeomorphism. In particular, for A’s in L with ∥A − 1∥ < \frac{1}{2}, there exist constants C_{1} and C_{2} so that

C_{1}∥A − 1∥_{2} ≤ ∥ |A| − 1∥_{2} ≤ C_{2}∥A − 1∥_{2}  (11.31)

Proof. By (11.30), A ↦ |A| is a smooth map.

The inverse map (strictly |A|^{2} ↦ A) is known as the Cholesky factorization; see [50, 109]. Given B > 0, apply the Gram–Schmidt procedure to the (linearly independent) columns of B working from right to left. This gives a factorization B = QA with Q unitary and A ∈ L. Note that |A| = B and that because the columns of B are linearly dependent, B ↦ A is also a smooth map. □

Theorem 11.12. Let J_{0} be a two-sided p-periodic Jacobi matrix with all gaps open and let Δ_{J_{0}} denote its discriminant. For a Jacobi matrix with parameters (a_{n}, b_{n}), the following are equivalent:

(i) Δ_{J_{0}}(J) − S^{p} − S^{−p} is a Hilbert–Schmidt operator on ℓ^{2}({0, 1, 2, …}).

(ii) ∑_{n} Tr{B_{n}^{2} + |A_{n} − 1|^{2}} < ∞.

(iii) ∑_{n} Tr{B_{n}^{2} + (|A_{n}| − 1)^{2}} < ∞.

(iv) ∑_{n} Tr{B_{n}^{2} + G(|A_{n}|)} < ∞.

(v) ∑_{m} d_{m}((a, b), T_{J_{0}})^{2} < ∞.

(vi) ∑_{m} d̃_{m}((a, b), T_{J_{0}})^{2} < ∞.

Here we have adopted the abbreviations A_{n} := A_{n,J_{0}}(J) and B_{n} := B_{n,J_{0}}(J)

Proof. (i)⇔(ii) amounts to the definition of the Hilbert–Schmidt norm. (ii)⇔(iii) follows from Proposition 11.11. (iii)⇔(iv) Notice that G, defined in (10.3), obeys

c_{ε}(x − 1)^{2} ≤ G(x) ≤ c′_{ε}(x − 1)^{2} ∀ x ∈ (ε, ε^{−1})

Applying this to the eigenvalues of |A_{n}| yields this equivalence.

(v)⇔(vi) is the q = 2 case of Proposition 3.5.

(vi)⇒(i) By Lemma 11.7, each matrix element of Δ_{J_{0}}(J)−S^{p} −S^{−p} is a smooth function of p consecutive pairs (a_{n}, b_{n}); moreover, by the magic formula, all of these smooth functions vanish if the corresponding p-tuple belongs to T_{J_{0}}. The implication now follows from the fact that smooth functions are Lipschitz.

(ii)⇒(v) Define J^{(k)} to be the p-periodic Jacobi matrix that equals J on block k, that is,

b_{ℓ}^{(k)} = b_{kp+ℓ} a_{ℓ}^{(k)} = a_{kp+ℓ} (11.32)

for ℓ = 1, 2, …, p. Obviously, J^{(k)} = J on block k and, by (11.29), the difference on blocks k − 1 and k + 1 are in ℓ^{2}, that is,

∑_{k}[ sup_{(k−1)p≤j≤(k+2)p−1} |b_{j} − b_{j}^{(k)}| + |a_{j} − a_{j}^{(k)}| ]^{2} < ∞  (11.33)

This implies (v) via Lemma 11.9. □

Proof of Theorem 1.4. We will refer to the three statements (i)–(iii) of Theorem 1.4 simply by their numbers.

Suppose first that

d_{m}((a, b), T_{J_{0}})) ∈ ℓ^{2}  (11.34)

Then (i) holds by Theorem 1.1. Moreover, by Theorem 11.12 and the hypothesis that all gaps are open, the RHS of (10.4) is finite. Therefore, the LHS is finite. Next we use this fact to prove (ii) and (iii).

As Δ′ is nonvanishing at all band edges,

∑_{j} F (Δ(E_{j})) < ∞ ⟺ ∑_{j=1}^{N} dist(E_{j}, σ_{ess}(J))^{3/2} < ∞  (11.35)

which verifies (iii). By Corollary 11.2,

Leftmost term in (10.4) < ∞ ⟺ (ii) holds  (11.36)

This completes the proof of (i)–(iii).

Conversely, if (i)–(iii) hold, then by (11.35), (11.36), and (10.4), we see that the RHS of (10.4) is finite. By Theorem 11.12, this implies (11.34). □

Proof of Theorem 1.3. Let β_{j} be the β’s associated to Δ(J), that is, |β_{j}| > 1, β_{j} +β_{j}^{−1} = E_{j} with E_{j} the eigenvalues of Δ(J) in ℝ\[−2, 2]. Then log|β_{j}| ∼ |β_{j}|−1 as β → ±1 small and |β_{j}| − 1 ∼ (|E_{j}| − 2)^{1/2}. Therefore,

(10.6) < ∞ ⟺ ∑_{j}(|E_{j}| − 2)^{1/2} ⟺ (1.24)  (11.37)

By Corollary 11.2,

(10.5) < ∞ ⟺ (1.25)  (11.38)

Finally, if {A_{n}, B_{n}}_{n=1}^{∞} are the p × p blocks in Δ(J), we have A_{n} = U|A_{n}| for some U with |det(U)| = 1, so

det|A_{n}| = |det(A_{n})| = ∏_{j=(n−1)p+1}^{np} ∏_{k=j}^{j+p=1}[\frac{a_{k}}{a_{k}^{(0)}} ]  (11.39)

by (3.3) and (2.14). Thus

∑_{n=1}^{N} log(det(|A_{n}|)) − p ∑_{k=1}^{Np} log[\frac{a_{k}}{a_{k}^{(0)}} ]

is bounded. Thus (1.27) is equivalent to (10.11).

By Theorem 10.2, we see that when (1.24) holds, then (1.25) ⇔ (1.27), and if they hold, (1.26) has a limit.

Moreover, if they hold, the hypotheses of Theorem 1.4 hold, so (1.28) is true. That (1.29) holds is a theorem of Peherstorfer–Yuditskii [84], but see the remark below. □

Remark. Using Theorem 6.8, we can provide a proof that {a_{n}}_{n=1}^{∞} has a periodic limit (of necessity, part of at least one element of T_{J_{0}}) without using the machinery of [84], but we have not succeeded in showing {b_{n}}_{n=1}^{∞} has a limit in this manner! Moreover, because of the current limitation of Theorem 6.8, we are restricted to requiring that {E_{j} : E_{j} ∈ σ(J_{0})} is finite. The proof is the following. Since L is a group, A_{1} … A_{n} is a lower triangular, so given via an adjoint of a QR algorithm. Let Ã_{j} be the equivalent type 2. Our A_{j} are type 3. By the continuity of the QR algorithm, A_{1} … A_{n} has a limit since Ã_{1} … Ã_{n} does. Looking at the first two diagonal elements, this implies that ∏_{n=1}^{mp} \frac{a_{j}}{a_{j}^{(0)}} and ∏_{n=2}^{mp+1} \frac{a_{j}}{a_{j}^{(0)}} have nonzero limits.

But their ratio is \frac{a_{mp+1}}{a_{1}} so a_{mp+1} has a limit. Similarly, a_{mp+ℓ} has a limit.

12. Szegő and Killip–Simon Theorems When Some Gaps Are Closed

Here we want to examine what might replace Theorems 1.3 and 1.4 if J_{0} is periodic but with some closed gaps. The Szegő-type theorem is almost the same as Theorem 1.3:

Theorem 12.1. Let J_{0} be any two-sided periodic Jacobi matrix with Jacobi parameters {a_{n}^{(0)}, b_{n}^{(0)}}_{n=−∞}^{∞}, and J a one-sided Jacobi matrix with Jacobi parameters {a_{n}, b_{n}}_{n=1}^{∞} and spectral measure dρ. Suppose that (1.22) holds, and that

∑_{m=1}^{N} dist(E_{m}, σ_{ess}(J))^{1/2} < ∞  (12.1)

if {E_{m}}_{m=1}^{N} is a labelling of the eigenvalues of J outside σ_{ess}(J). Then

− ∫_{σ_{ess}(J_{0})} log(\frac{dρ_{ac}}{dx} )dist(x, ℝ \ σ_{ess}(J_{0}))^{−1/2} dx < ∞  (12.2)

implies

lim( ∑_{j=1}^{pN} log(\frac{a_{j}}{a_{j}^{(0)}} ))  (12.3)

exists and lies in (−∞, ∞). Conversely, (12.2) holds so long as

lim sup( ∑_{j=1}^{N} log(\frac{a_{j}}{a_{j}^{(0)}} )) > −∞  (12.4)

and then the limit in (12.3) exists and lies in (−∞, ∞).

Moreover, if (12.2) or (12.4) holds, then there is J_{1} ∈ T_{J_{0}}, so

d_{m}(J, J_{1}) → 0  (12.5)

Remark. All that is missing is (1.28) which we do not claim. However, since (12.1)/(12.2) imply (i)–(iii) of Theorem 12.3 below, we have (12.7).

Proof. As noted, even with closed gaps, (1.25) is equivalent to (11.10) (see Corollary 11.2). Once one notes this, the proof of Theorem 1.3 provides all the results stated as Theorem 12.1. □

Theorem 1.4 used open gaps in two ways. First, in the translation of a matrix pseudo-Szegő condition, (11.10) with α = \frac{1}{2} to the original spectral measure of J, and second, translating a Hilbert–Schmidt bound on Δ(J) − S^{p} − S^{−p} to ℓ^{2} approach to the isospectral torus. The second issue can be finessed if we leave things as a Hilbert–Schmidt condition, which reduces to a sum of translates of an explicit positive polynomial in the a_{n}’s and b_{n}’s being finite. As for translating (11.10) with α = \frac{1}{2}, the argument that proved Corollary 11.2 translates immediately to

Lemma 12.2. Suppose σ(J_{0}) has closed gaps at {y_{j}}_{j=1}^{ℓ} ⊂ σ(J_{0}). Then (11.10) holds with α = \frac{1}{2} if and only if

∫_{σ_{ess}(J_{0})} dist(x, ℝ \ σ(J_{0}))^{1/2} ∏_{j=1}^{ℓ}|x − y_{j}|^{2}|log(ω(x))| dx < ∞  (12.6)

Plugging this into our proof of Theorem 1.4 immediately yields

Theorem 12.3. Let J_{0} be a two-sided periodic Jacobi matrix with closed gaps at {y_{j}}_{j=1}^{ℓ} ⊂ σ(J_{0}) and let J be a Jacobi matrix. Then

Tr((Δ_{J_{0}}(J) − S^{p} − S^{−p})^{2}) < ∞  (12.7)

if and only if

(i) (1.22) holds. (ii) (1.24) holds with \frac{1}{2} replaced by \frac{3}{2}.

(iii) (12.6) holds.

Example 12.4. Take J_{0} to be the two-sided free Jacobi matrix but think of it as period 2. Then

Δ_{J}(x) = x^{2} − 2

and a direct calculation of J^{2} − S^{2} − S^{−2} shows that (12.7) is equivalent to the three conditions

∑_{n}(a_{n}^{2} + b_{n}^{2} + a_{n+1}^{2} − 2)^{2} < ∞  (12.8)
∑_{n}(a_{n+1}(b_{n} + b_{n+1}))^{2} < ∞  (12.9)
∑_{n}(a_{n}a_{n+1} − 1)^{2} < ∞  (12.10)

If b_{n} = 0 and

a_{n} = 1 + (−1)^{n}(n + 1)^{−β}  (12.11)

then (12.8)–(12.10) hold if and only if β > \frac{1}{4} while, of course, (1.13) requires β > \frac{1}{2}. This is one of many known extensions of the (J_{0}-free) Killip–Simon theorem (see, e.g., Laptev et al. [65], Kupin [64], and Nazarov et al. [78]). Some of these results have MOPRL analogs which, via the magic formula, lead to variants of Theorems 1.4 and 12.3.

13. Eigenvalue Bounds for MOPRL

There are Birman–Schwinger kernels for MOPRL and it should be possible to extend the proofs of most bounds on the number of eigenvalues outside [−2, 2] or on moments of |E_{j}| − 2 from the scalar to matrix case with optimal constants. But if one is willing to settle for less than optimal constants (but still not awful constants), there is a simple method to go from the scalar to matrix case. It depends on the following:

Theorem 13.1. Let J be an ℓ×ℓ block Jacobi matrix in the Nevai class with Jacobi parameters {A_{n}, B_{n}}_{n=1}^{∞}. Let E_{j}^{±}(J) denote its eigenvalues counting multiplicity outside [−2, 2], that is, E_{1}^{+} ≥ E_{2}^{+} ≥ ⋯ > 2 > −2 > ⋯ ≥ E_{2}^{−} ≥ E_{1}^{−}. Let J_{±} be the scalar Jacobi matrix with a_{n} ≡ 1 and

b_{n}^{±} = ±∥B_{n}∥ ± ∥A_{n−1} − 1∥ ± ∥A_{n} − 1∥  (13.1)

and let J_{±}^{(ℓ)} be an ℓ-fold direct sum of J_{±}. Then

|E_{j}^{±}(J)| ≤ |E_{j}^{±}(J_{±}^{(ℓ)})|  (13.2)

Proof. The matrix analog of the observation of Hundertmark–Simon [53] extended to 2ℓ × 2ℓ matrices (with ℓ × ℓ blocks) says that

(\begin{matrix}−|A_{n} − 1| & 1 \\ 1 & −|A_{n} − 1| \end{matrix}) ≤ (\begin{matrix}0 & A_{n} \\ A_{n}^{∗} & 0 \end{matrix}) ≤ (\begin{matrix}|A_{n} − 1| & 1 \\ 1 & |A_{n} − 1| \end{matrix})  (13.3)

since

∥(\begin{matrix}0 & C \\ C^{∗} & 0 \end{matrix})∥^{2} = ∥(\begin{matrix}C^{∗}C & 0 \\ 0 & CC^{∗} \end{matrix})∥ = ∥C∥^{2}  (13.4)

Thus writing J_{±}^{(ℓ)} as ℓ × ℓ blocks with each block a multiple of 1,

J_{−}^{(ℓ)} ≤ J ≤ J_{+}^{(ℓ)}  (13.5)

from which (13.2) is immediate. □

Corollary 13.2. For any block Jacobi matrix, J, in Nevai class,

∑_{j,±} (E_{j}^{±}(J)^{2} − 4)^{1/2} ≤ 2ℓ ∑_{n} ∥B_{n}∥ + 4ℓ ∑_{n} ∥A_{n} − 1∥  (13.6)

Remark. In particular, this implies if the RHS of (13.6) is finite, so is the LHS.

Proof. Hundertmark–Simon [53] proved

∑_{j} (E_{j}^{±}(J_{±})^{2} − 4)^{1/2} ≤ ∑_{n} b_{n}^{±}  (13.7)

from which (13.6) follows by (13.2). □

14. The Analog of Nevai’s Conjecture

Proof of Theorem 1.5. By (1.32), J −J_{0} is trace class. Thus J^{ℓ}−J_{0}^{ℓ} = ∑_{k=0}^{ℓ−1} J^{k}(J − J_{0})J^{ℓ−1−k} is trace class, so Δ_{J_{0}}(J) − Δ_{J_{0}}(J_{0}) = Δ_{J_{0}}(J) − (S^{p} + S^{−p}) is trace class.

It follows that if (1.32) holds and Δ(J) has matrix Jacobi parameters {A_{n}, B_{n}}_{n=1}^{∞} that

∑_{n=1}^{∞} ∥B_{n}∥ + ∑_{n=1}^{∞} ∥A_{n} − 1∥ < ∞  (14.1)

By Corollary 13.2, the eigenvalues Δ(J) obey

∑_{j=1}^{∞} (|E_{j}^{±}| − 2)^{1/2} < ∞  (14.2)

(14.1) also implies

∑_{n=1}^{∞} |log(det|A_{n}|)| < ∞  (14.3)

We can apply Theorem 10.2 and conclude that Z(J) is finite, that is,

∫ (4 − E^{2})^{−1/2} log(det(W (E))) dE > −∞  (14.4)

By Corollary 11.2, we obtain (1.25). □

15. Perturbations of Periodic OPUC

In this final section, we want to present the translations of our results to OPUC. Since the magic formula maps periodic OPUC to MOPRL, the changes needed in the proofs will be minor, although for the analog of Theorem 1.4, there is one significant change. It is interesting to note the sequence of mappings for the OPUC periodic Rakhmanov’s theorem. We map OPUC to MOPRL using the magic formula and them map that to MOPUC using the Szegő map.

The OPUC version of Theorem 1.1 is already in Last–Simon [66]. As for Theorem 1.2:

Theorem 15.1. Let C_{0} be a two-sided periodic CMV matrix. Let C be an ordinary CMV matrix with Verblunsky coefficients {α_{n}}_{n=0}^{∞}. Suppose

Σ_{ac}(C) = σ_{ess}(C_{0})  (15.1)

Then, as m → ∞,

d_{m}(α, T_{C_{0}}) → 0  (15.2)

Proof. Let C_{r} be a right limit of C. By Theorem 7.1, Δ(C_{r}) = S^{p} + S^{−p}. Thus, by Theorem 4.1, C_{r} ∈ T_{C_{0}}. □

As for the analog of Theorem 1.3, if we drop discussion of ℓ^{2} convergence, it holds, similar to Theorem 12.1.

Theorem 15.2. Let C_{0} be a two-sided periodic CMV matrix with Verblunsky coefficients {α_{j}^{(0)}}_{j=−∞}^{∞}, and C a one-sided CMV matrix with Verblunsky coefficients {α_{j}}_{j=0}^{∞} and spectral measure dμ. Suppose that σ_{ess}(C) = σ(C_{0}) and

∑_{m=1}^{N} dist(E_{m}, σ_{ess}(C))^{1/2} < ∞  (15.3)

where {E_{m}}_{m=1}^{N} is a labelling of the eigenvalues of C outside σ_{ess}(C_{0}). Then

− ∫_{σ_{ess}(C_{0})} log(\frac{dμ_{ac}}{dθ} )dist(θ, ℝ \ σ_{ess}(C_{0}))^{−1/2} \frac{dθ}{2π} < ∞  (15.4)

implies

lim_{N→∞}( ∑_{j=1}^{pN} log(\frac{ρ_{j}}{ρ_{j}^{(0)}} ))  (15.5)

exists and lies in (−∞, ∞). Conversely, (1.25) holds so long as

lim sup_{N→∞}( ∑_{j=1}^{N} log(\frac{ρ_{j}}{ρ_{j}^{(0)}} )) > −∞  (15.6)

and then the limit in (15.5) exists and lies in (−∞, ∞).

Remarks. 1. We have not stated that the α_{n} have a limit in T_{C_{0}}. We suspect that the methods of [84] extend to the OPUC but have not checked this and they do not explicitly mention it.

2. Of course, if this theorem is applicable and C_{0} obeys the conditions of Theorem 15.3 below, then we have a result on ℓ^{2} convergence to T_{C_{0}}.

3. One can replace ρ_{j}^{(0)} by the logarithmic capacity of σ(C_{0}).

Proof. At open gaps, Δ′(e^{iθ}) ≠ 0, so (15.3) is equivalent to

∑_{\begin{matrix}E∉[−2,2] \\ E∈σ(Δ(C)) \end{matrix}} (|E| − 2)^{1/2} < ∞  (15.7)

Moreover, by (4.4)–(4.7), we have that

log(det|A_{n}|) = log(det(A_{n})) = log[ ∏_{j=(n−1)p+1}^{np} ∏_{k=j}^{j+p^{−1}} \frac{ρ_{k}}{ρ_{k}^{(0)}} ]  (15.8)

Now just follow the proof of Theorem 12.1. □

In carrying over Theorem 1.4 to OPUC, one runs into a serious roadblock: van Moerbeke’s theorem [108] that the Hamiltonian flows generated by the coefficients of the t_{j} (given by (11.17)) are independent is not known for OPUC. Instead, we use a weaker result of Simon [99, Section 11.10] that proves the derivatives of coefficients of Δ span the normal bundle for a dense open set that is ‘most’ of the points with all gaps open. We will call the isospectral tori in this dense open set the generic independence tori. Then by mimicking the arguments in Section 11, we obtain

Theorem 15.3. Let C_{0} be a two-sided CMV matrix in a generic independence torus with Verblunsky coefficients {α_{j}^{(0)}}_{j=−∞}^{∞}. Let C be a CMV matrix with Verblunsky coefficients {α_{j}}_{j=0}^{∞}. Then

∑_{m=0}^{∞} d_{m}(α, T_{C_{0}})^{2} < ∞  (15.9)

if and only if

(i) σ_{ess}(C) = σ(C_{0}) (ii) For the eigenvalues {E_{j}}_{j=1}^{N} not in σ(C_{0}),

∑_{j=1}^{N} dist(E_{j}, σ(C_{0}))^{3/2} < ∞  (15.10)

(iii) If μ is the spectral measure for C, then

− ∫_{σ(C_{0})} log(\frac{dμ_{ac}}{dθ} )dist(θ, ∂𝔻 \ σ_{ess}(C_{0}))^{1/2} \frac{dθ}{2π} < ∞  (15.11)

Our last result is a periodic OPUC version of Nevai’s conjecture.

Theorem 15.4. Let C_{0} be a two-sided p-periodic CMV matrix and let C be a CMV matrix with Verblunsky coefficients {α_{j}}_{j=0}^{∞}. Then

∑_{m=0}^{∞} d_{m}(α, T_{C_{0}}) < ∞  (15.12)

implies (15.4)

All the above results assume the period p is even. However, by using sieving (see Example 1.6.14 of [98]) to map period p to period 2p, one can extend Theorems 15.1 and 15.2 to p odd. In particular, we obtain the p = 1 results of [10, 2, 9] as very special cases of ours.

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