ZEROS OF OPUC AND LONG TIME ASYMPTOTICS OF SCHUR AND RELATED FLOWS

Barry Simon

Mathematics 253-37 California Institute of Technology Pasadena, CA 91125, USA

Abstract. We provide a complete analysis of the asymptotics for the semi-infinite Schur flow: α_{j}(t) = (1 − |α_{j}(t)|^{2})(α_{j+1}(t) − α_{j−1}(t)) for α_{−1}(t) = 1 boundary conditions and n = 0, 1, 2, …, with initial condition α_{j}(0) ∈ (−1, 1). We also provide examples with α_{j}(0) ∈ 𝔻 for which α_{0}(t) does not have a limit. The proofs depend on the solution via a direct/inverse spectral transform.

1. Introduction

One of our purposes in this paper is to study the long time asymptotics of the solution of the differential equation on (−1, 1)^{∞} for t ≥ 0, n ≥ 0,

α′_{n}(t) = (1 − |α_{n}(t)|^{2})(α_{n+1}(t) − α_{n−1}(t))  (1.1)

where α_{−1} is interpreted as

α_{−1}(t) = −1  (1.2)

for arbitrary boundary conditions α_{n}(0) ∈ (−1, 1). This is called the Schur flow [1, 2, 8, 11]. We will also say something about complex initial conditions with α_{n}(t) ∈ 𝔻 = {z | |z| < 1}.

We got interested in this problem due to work of Golinskii [11] who proved for initial conditions α_{n}(0) = 0 (all n ≥ 0) that α_{n}(t) → (−1)^{n}, and he obtains the leading O(1/t) correction. From this point of view, our main result is the following:

Theorem 1.1. Suppose each α_{n}(0) ∈ (−1, 1). Then one of the following holds: (i)

α_{n}(t) → (−1)^{n}  for all n  (1.3)

1991 Mathematics Subject Classification. 05E35, 33D45, 37K10.

Key words and phrases. Schur flows, orthogonal polynomials, Toda flows.

∗ Supported in part by NSF grant DMS-0140592 and U.S.–Israel Binational Science Foundation (BSF) Grant No. 2002068.

(ii) There exists 1 ≤ N ≤ ∞ and

1 > x_{1} > x_{2} > ⋯ > x_{N} > −1  (1.4)

so that

\begin{matrix}α_{2n}(t) → 1 n ≥ 0,  α_{2n−1} → −x_{n} all 1 ≤ n < N \\ α_{2n−1} → −x_{N} all n ≥ N \end{matrix}  (1.5)

(iii) There exists 1 ≤ N ≤ ∞ and x_{j} obeying (1.4) so that

\begin{matrix}α_{2n+1}(t) → −1 n ≥ 0,  α_{2n}(t) → x_{n+1} 0 ≤ n < N \\ α_{2n}(t) → x_{N} n ≥ N − 1 \end{matrix}  (1.6)

We will have a complete spectral theory analysis of which case one has based on the initial conditions. But for now, we note:

Proposition 1.2. (i) If α_{n}(0) → 0 as n → ∞, then we are in case

(i) of Theorem 1.1. (ii) If (−1)^{n}α_{n}(0) → ±1 as n → ∞, then we are in case (i) of Theo-

rem 1.1. (iii) If 0 < a < 1 and α_{n}(0) → a as n → ∞, then we are in case (ii) of

Theorem 1.1 and

x_{N} = 1 − 2a^{2} if N < ∞ or x_{n} ↓ 1 − 2a^{2} if N = ∞  (1.7)

(iv) If −1 < a < 0 and α_{n}(0) → a as n → ∞, then we are in case (iii)

of Theorem 1.1 and (1.7) holds.

(v) Case (ii) holds with x_{n} ↓ −1 as n → ∞ if and only if α_{n}(0) → 1 as n → ∞.

(vi) Case (iii) holds with x_{n} ↓ −1 as n → ∞ if and only if α_{n}(0) → −1 as n → ∞.

Note. These possibilities are consistent with the partial results of Theorem 5 of Golinskii [11].

The situation is rather more subtle if we allow complex initial conditions:

Proposition 1.3. There exists {α_{n}(0)} ∈ 𝔻^{∞} so that α_{0}(t) does not have a limit.

Besides this, we will discuss rates of convergence. In cases (ii) and (iii) for j < 2N+1 (resp. j < 2N), the rate will be exponentially fast. In other cases, the situation can be subtle, although if ∑_{n=0}^{∞}|α_{n}(0)| < ∞, the rate is that found by Golinskii when α_{n}(0) = 0.

At first glance, it seems surprising that one can obtain such detailed information for a nonlinear equation. The reason, of course, is that (1.1) is completely integrable. Indeed, it is exactly solvable via a spectral transform [8, 12, 11]. In this sense, this problem is a close analog of work of Moser [14, 15] and Deift–Li–Tomei [5] on asymptotics of Toda flows, except for an extra subtlety we will discuss shortly.

Just as Toda is closely connected to the theory of orthogonal polynomials on the real line (OPRL), the theory of orthogonal polynomials on the unit circle (OPUC) [21, 10, 17, 18, 16] will be central here. As Golinskii [11] notes for Toda, OPRL “plays one of the first fiddles in the performance (albeit not entering the final results directly).” To push his metaphor, the present paper promotes OPUC and OPRL to concert soloist—OPs enter directly into the results (see Theorem 1.5 below), and more directly in our proofs than in previous works.

Recall (e.g., [16, 17]) that nontrivial probability measures, dμ, on ∂𝔻 = {z | |z| = 1} are parametrized by {α_{n}}_{n=0}^{∞} ∈ 𝔻^{∞} via the Szegő recursion relations

Φ_{n+1}(z) = zΦ_{n}(z) − ᾱ_{n}Φ_{n}^{∗}(z)  (1.8)
Φ_{n}^{∗}(z) = z^{n} \overline{Φ_{n}(1/z̄)}  (1.9)

where Φ_{n}(z) are the monic orthogonal polynomials in L^{2}(∂𝔻, dμ). We often use α_{n}(dμ) when we want to make the dμ-dependence explicit. The situation of (1.1) is made explicit by

Proposition 1.4 ([8, 11]; also see our first appendix and [4]). Given α_{n}(0) ∈ 𝔻^{∞}, define dμ by

α_{n}(dμ) = α_{n}(0)  (1.10)

and dμ_{t} by

dμ_{t}(θ) = \frac{e^{2t cos(θ)} dμ(θ)}{∫ e^{2t cos(θ)} dμ(θ)}  (1.11)

Then

α_{n}(t) ≡ α_{n}(dμ_{t})  (1.12)

is the unique solution of (1.1) with initial conditions α_{n}(0).

Following the analog of Deift–Li–Tomei for Toda flows [5], we will want to consider generalized Schur flows associated to any bounded real-valued function G(θ) on ∂𝔻 via

Σ_{t}(dμ)(θ) = \frac{e^{tG(θ)} dμ(θ)}{∫ e^{tG(θ)} dμ(θ)}  (1.13)
σ_{t}(α_{n}(0)) = α_{n}(Σ_{t}(dμ))  (1.14)

When we want to make G explicit, we will use σ_{t}^{G} and Σ_{t}^{G}. We will say more about these flows in the first appendix. [12] calls these generalized Ablowitz–Ladik flows. Since generalized Schur flows preserve reality only if G(−θ) = G(θ), there is some reason in their choice, but we prefer to emphasize the connection to Schur functions and OPUC.

One can also define Σ_{t} and σ_{t} for trivial measures, that is, measures on ∂𝔻 with finite support

dμ = ∑_{j=1}^{m} μ_{j}δ_{e^{iθ_{j}}}  (1.15)

parametrized by {α_{j}}_{j=0}^{m−1} with α_{0}, …, α_{m−2} ∈ 𝔻 and α_{m−1} = (−1)^{m+1} ∏_{j=1}^{m} e^{iθ_{j}}. In this case, since all that matters are {G(θ_{j})}_{j=1}^{m}, we can suppose G is a polynomial. For this case, the asymptotics of generalized Schur flows were studied by Killip–Nenciu [12].

We will also analyze the long time asymptotics in this case. To do so, we will let

z_{j} = e^{iθ_{j}}  (1.16)

and renumber, so

G(z_{1}) ≥ G(z_{2}) ≥ ⋯ ≥ G(z_{m})  (1.17)

We define the K-groups K_{1}, …, K_{ℓ} to be those indices K_{1} ≡ {1, …, k_{1}}, K_{2} ≡ {k_{1} + 1, … k_{2}}, …, K_{ℓ} = {k_{ℓ−1} + 1, …, k_{ℓ} ≡ m}, so G(z_{j}) = G(z_{p}) if j, p ∈ K_{ℓ} and so G(z_{k_{j}}) > G(z_{k_{j}+1}). Thus, the K-groups are the level sets of G on {z_{j}}_{j=1}^{m}.

It will also be convenient to define

z^{(K_{j})} = ∏_{p≤k_{j−1}} z_{j}  (1.18)

the product of those z where G is larger than the value common in K_{j}. Also, given an initial point measure, dμ of the form (1.15), we define the K-group induced measure by

dμ^{(K_{j})} = ∑_{ℓ=k_{j−1}+1}^{k_{j}} μ_{ℓ}^{(K_{j})}δ_{e^{iθ_{ℓ}}}  (1.19)

where

μ_{ℓ}^{(K_{j})} = \frac{[∏_{p=1}^{k_{j−1}}|z_{ℓ} − z_{p}|^{2}]μ_{ℓ}}{∑_{m=k_{j−1}+1}^{k_{j}} [∏_{p=1}^{k_{j−1}}|z_{m} − z_{p}|^{2}]μ_{m}}  (1.20)

Then we will prove:

Theorem 1.5. Let dμ be a finite measure given by (1.15) and G defined on {z_{j}}_{j=1}^{m} and real-valued. Then for ℓ ∈ K_{j},

σ_{t}(α_{ℓ−1}) → (−1)^{k_{j−1}} \overline{z^{(K_{j})}} α_{ℓ−k_{j−1}−1}(dμ^{(K_{j})})  (1.21)

Killip–Nenciu also obtain a limit theorem; we will discuss its relation to ours in Section 3.

At first sight, the problem looks very easy. α_{n}(·) is continuous under weak convergence (a.k.a. vague or weak-∗) convergence of measures, so one need only find the weak limit of dμ_{t} or Σ_{t}(dμ). For the Schur flow case, dμ_{t} has either a one- or two-point support for its weak limit. But for trivial (i.e., finite point) limits, lim α_{n}(·) is only determined for n smaller than the number of points in the support of the limit, so this only determines at most two α’s! The issue can be seen clearly in the context of Theorem 1.5: dμ_{t} converges to dμ^{(K_{1})} and so only determines {α_{ℓ−1}(∞)}_{ℓ=1}^{k_{1}}.

The key to the general analysis is to track the zeros, {z_{j}^{(n)}}_{j=1}^{n}, of the OPUC, Φ_{n}(z). By (1.8), Φ_{n}^{∗}(0) = 1 and

Φ_{n}(z) = ∏_{j=1}^{n}(z − z_{j}^{(n)})  (1.22)

so we have

α_{n} = (−1)^{n} ∏_{j=1}^{n+1} \overline{z_{j}^{(n+1)}}  (1.23)

The zeros in turn are determined by the Szegő variational principle, that

∫ | ∏_{j=1}^{n}(z − w_{j})|^{2} dμ(θ)  (1.24)

is minimized precisely with {w_{j}}_{j=1}^{n} = {z_{j}^{(n)}}_{j=1}^{n}.

We will find a consequence of the variational principle (Theorem 3.4 below) that lets us use weak convergence beyond the naive limit.

These same ideas work for the Toda flows and are in some ways simpler there since the exponential factor there, e^{2tx}, is strictly monotone. We begin in Section 2, as a warmup, by proving Moser’s theorem on the asymptotics for finite Toda flows. In Section 3, we prove Theorem 1.5. In Section 4, we prove Theorem 1.1 and Proposition 1.2—as in Deift–Li–Tomei, the extreme points in the essential spectrum are crucial. Section 5 discusses second-order corrections. In Section 6, we present the example of Proposition 1.3, and in Section 7, discuss hypotheses which prevent the pathologies there. Appendix A combines ideas of Deift–Li–Tomei [5, 6] for Toda flows with notions from Killip–Nenciu [12] to talk about difference equations associated with (1.13). Appendix B provides a new proof and strengthening of results of Denisov–Simon on zeros of OPUC near isolated points of supp(dμ).

We should note that while we show the asymptotics of α_{n}(t) for n fixed and t → ∞ is simple, and one can see that, often, asymptotics of α_{n}(t) for t fixed and n → ∞ is easy (e.g., Golinskii [11] shows that α_{n}(0) ∈ ℓ^{1} (resp. ℓ^{2}) implies α_{n}(t) ∈ ℓ^{1} (resp. ℓ^{2})), the subtle asymptotics is for α_{n}(t) as t → ∞ and n/t → q ∈ (0, ∞). For α_{n}(0) ≡ 0, this is studied using Riemann–Hilbert methods in [3], and no doubt their methods extend to any case |α_{n}(0)| ≤ e^{−cn} for c > 0. Indeed, using ideas of [13], one can probably handle other classes where dμ is not analytic.

2. Moser’s Theorem on Toda Asymptotics

As a warmup, we consider a probability measure

dρ(x) = ∑_{j=1}^{N} ρ_{j}δ_{x_{j}}  (2.1)

where each ρ_{j} > 0, and the family of measures

dρ_{t}(x) = \frac{e^{2tx} dρ(x)}{∫ e^{2tx} dρ(x)}  (2.2)

The Jacobi parameters, {a_{n}}_{n=1}^{N−1} ∪{b_{n}}_{n=1}^{N}, associated to dρ are defined by looking at the recursion relations associated to the monic orthogonal polynomials

xP_{n}(x) = P_{n+1}(x) + b_{n+1}P_{n}(x) + a_{n}^{2}P_{n−1}(x)  (2.3)

with a_{n} > 0. The Jacobi parameters associated to dρ_{t} obey the Toda equations in Flaschka form:

\frac{da_{n}}{dt} = a_{n}(b_{n+1} − b_{n})  (2.4)
\frac{db_{n}}{dt} = 2(a_{n}^{2} − a_{n−1}^{2})  (2.5)

(with a_{0} = a_{N} ≡ 0 in (2.5)).

We order the x_{j}’s by

x_{1} > x_{2} > ⋯ > x_{N}

Then our main result in this section is the following:

Theorem 2.1. The above finite Toda chains obey

lim_{t→∞} b_{j}(t) = x_{j}  lim_{t→∞} a_{j}(t) = 0  (2.6)

(the first for j = 1, …, N and the second for j = 1, 2, …, N − 1). Moreover, errors for b_{n} are O(e^{−ct}) and for a_{n} are O(e^{−ct/2}) where

c = min_{j=1,2,...,N−1}(x_{j} − x_{j+1})  (2.7)

Remarks. 1. This is a celebrated result of Moser [14], but it is interesting to see it proven using zeros of OPRL. In any event, it is a suitable warmup for our result on OPUC.

2. The O(e^{−ct}) and O(e^{−ct/2}) estimates are not ideal; we will discuss this further and obtain finer error estimates at the end of this section.

3. The same proof shows that as t → −∞, b_{j} → x_{N+1−j} and a_{j} → 0.

P_{j}(x) has j simple zeros x_{1}^{(j)} > x_{2}^{(j)} > ⋯ > x_{j}^{(j)}. (It is known that x_{k} > x_{k}^{(j+1)} > x_{k}^{(j)} > x_{n−j+k−1}, but we won’t need that.) The key fact is:

Theorem 2.2. For each j = 1, …, N and k = 1, …, j,

lim_{t→∞} x_{k}^{(j)}(t) = x_{k}  (2.8)

The errors are O(e^{−ct}) with c given by (2.7).

Remark. Since P_{N}(x) = ∏_{j=1}^{N}(x − x_{j}), we have x_{j}^{(N)} = x_{j} for all t.

Proof. P_{j}(x) is the projection of x^{j} on {1, …, x^{j−1}}^{⊥}, so for any monic polynomial Q(x) of degree j,

∫ |Q(x)|^{2} dρ(x) ≥ ∫ |P_{j}(x)|^{2} dρ(x)  (2.9)

Pick Q(x) = ∏_{ℓ=1}^{j}(x − x_{ℓ}), which minimizes the contributions of x_{1}, …, x_{j} to the integral, and see that

∫ |P_{j}(x)|^{2} dρ(x) ≤ e^{2tx_{j+1}}(x_{1} − x_{N})^{2j}  (2.10)

since |Q(x_{ℓ})| ≤ |x_{ℓ} − x_{1}|^{j} for ℓ ≥ j + 1. On the other hand,

|P_{j}(x)| ≥ [ min_{ℓ=1,...,j} |x − x_{ℓ}^{(j)}|]^{j}  (2.11)

so for q = 1, …, j,

ρ_{q}e^{2tx_{q}} min_{ℓ=1,...,j} [|x_{q} − x_{ℓ}^{(j)}|^{j}]^{2} ≤ ∫ |P_{j}(x)|^{2} dρ(x)  (2.12)

We conclude for q ≤ j that

min_{ℓ=1,...,j} |x_{q} − x_{ℓ}^{(j)}|^{2j} ≤ ρ_{q}^{−1}(x_{1} − x_{N})^{2j}e^{2t(x_{j+1}−x_{q})}  (2.13)

which shows that each x_{q} has an x_{ℓ}^{(j)} exponentially near to it, but only O(e^{−ct/j}).

But once we know each such x_{q} has one zero exponentially near, we see that for t large, all other zeros are a distance at least \frac{1}{2}c away. Thus (2.11) can be replaced, for t large, by

|P_{j}(x)| ≥ (\frac{c}{2})^{j−1} min_{ℓ=1,...,j} |x − x_{ℓ}^{(j)}|  (2.14)

Plugging this into (2.12) and finding the analog of (2.13) leads to an O(e^{−ct}) error. Explicitly, (2.13) is replaced by

min_{ℓ=1,...,j} |x_{q} − x_{ℓ}^{(j)}|^{2} ≤ Ce^{2t(x_{j+1}−x_{q})}  (2.15)
□

Remark. We will need a better error estimate in the next section and show how to get it later in the section.

Proof of Theorem 2.1. (2.3) can be rewritten:

∏_{ℓ=q}^{n+1}(x − x_{j}^{(n+1)}) = (x − b_{n+1}) ∏_{ℓ=1}^{n}(x − x_{j}^{(n)}) − a_{n}^{2}P_{n−1}(x)  (2.16)

Identifying the x^{n} and x^{n−1} terms, we see the analog of (1.23); the first for n = 1, 2, …, N and the second for n = 1, 2, …, N − 1:

b_{n+1} = ∑_{j=1}^{n+1} x_{j}^{(n+1)} − ∑_{j=1}^{n} x_{j}^{(n)}  (2.17)

a_{n}^{2} = b_{n+1} ∑_{j=1}^{n} x_{j}^{(n)} + ∑_{1≤j<ℓ≤n} x_{j}^{(n)}x_{ℓ}^{(n)} − ∑_{1≤j<ℓ≤n+1} x_{j}^{(n+1)}x_{ℓ}^{(n+1)} (2.18)

(2.8) and the error estimates of Theorem 2.2 immediately imply b_{j}(t)− x_{j} = O(e^{−ct}) and a_{j}(t)^{2} = O(e^{−ct}). □

We next want to note, following Moser, that the differential equations (2.4) and (2.5) yield better error estimates than Theorem 2.1 has and then explain how to improve the estimates on zeros to get better estimates on the errors of b_{n} and a_{n} with the zeros.

Once we know b_{j}(t) → x_{j}, (2.2) implies

t^{−1} log a_{j}(t) → x_{j+1} − x_{j} < 0  (2.19)

Indeed, since the approach of b_{j}(t) to x_{j} is exponentially fast,

log a_{j}(t) − t(x_{j+1} − x_{j}) → log C_{j}  (2.20)

for some finite C_{j}, and thus,

a_{j} ∼ C_{j}e^{−t(x_{j+1}−x_{j})}  (2.21)

proving the error should be O(e^{−ct}), not O(e^{−ct/2}). Then plugging (2.21) into (2.5), we see that

|b_{j}(t) − x_{j}| ≤ C̃_{j} exp(−2t min[(x_{j+1} − x_{j}), (x_{j} − x_{j−1})])  (2.22)

with the right side being the exact order of error if x_{j+1}−x_{j} ≠ x_{j} −x_{j−1} (if there is equality, a_{j}^{2} and a_{j−1}^{2} can completely or partially cancel). Thus, the error is O(e^{−2ct}), not O(e^{−ct}).

To improve our estimates on zeros, we use the minimization principle to get a self-consistency equation on the zeros. This result, proven using orthogonality of P to P/(x − x_{0}) is well-known (see (3.3.3) of [21]); we give a variational principle argument in line with the strategy in this paper. The OPUC analog is (1.7.51) of [17].

Lemma 2.3. The zeros x_{k}^{(j)} of P_{j}(x) obey

x_{k}^{(j)} = \frac{∫ x ∏_{ℓ≠k}|x − x_{ℓ}^{(j)}|^{2} dρ(x)}{∫ ∏_{ℓ≠k}|x − x_{ℓ}^{(j)}|^{2} dρ(x)}  (2.23)

In particular, for any y,

|x_{k}^{(j)} − y| ≤ \frac{∫ |x − y| ∏_{ℓ≠k}|x − x_{ℓ}^{(j)}|^{2} dρ(x)}{∫ ∏_{ℓ≠k}|x − x_{ℓ}^{(j)}|^{2} dρ(x)}  (2.24)

Proof. Since ∫ ∏_{n=1}^{j}|x − y_{ℓ}|^{2} dρ(x) is minimized at y_{ℓ} = x_{ℓ}^{(j)}, the derivative with respect to y_{k} at this point is zero, that is,

∫ (x − x_{k}^{(j)}) ∏_{ℓ≠k}|x − x_{ℓ}^{(j)}| dρ(x) = 0

which is (2.23). (2.23) implies (2.24) by noting x_{k}^{(j)} − y is given by (2.23) with the first x in the integrand replaced by x − y. □

Theorem 2.4. For j = 1, …, N and k = 1, …, j,

|x_{k}^{(j)}(t) − x_{k}| ≤ Ce^{−2t(x_{k}−x_{j+1})}  (2.25)

Proof. We begin by noting that since the {x_{ℓ}^{(j)}(t)}_{ℓ≠k} for t large are very near x_{ℓ} (and so, not near x_{k}) that for some T_{0} and constant C_{1} and all t ≥ T_{0},

∫ ∏_{ℓ≠k} |x − x_{ℓ}^{(j)}|^{2} dρ_{t}(x) ≥ C_{1}e^{2tx_{k}}  (2.26)

Moreover, since x_{1} > x_{2} > …,

∑_{m≥j+1} ρ_{m}e^{2tx_{m}}|x_{m} − x_{k}| ∏_{ℓ≠k} |x_{m} − x_{ℓ}^{(j)}|^{2} ≤ Ce^{2tx_{j+1}}  (2.27)

so we need only control the terms m = 1, 2, …, j in estimating (2.24). For m = 1, …, j, we use (2.15) to see

ρ_{n}e^{2tx_{m}}|x_{m} − x_{k}| ∏_{ℓ≠k} |x_{m} − x_{ℓ}^{(j)}|^{2} ≤ Ce^{2tx_{m}}e^{2t(x_{j+1}−x_{m})}
≤ Ce^{2tx_{j+1}}

Thus all terms in the numerator of (2.24) with y = x_{k} are bounded by Ce^{2tx_{j+1}}. Combining this with (2.26), we obtain (2.25). □

Remark. Putting the improved bound (2.25) in place of (2.15) shows that the sum in (2.27) dominates the sum in the numerator of (2.24).

Theorem 2.5. We have

|b_{j}(t) − x_{j}| ≤ C_{3}[exp(2t(x_{j+1} − x_{j})) + exp(2t(x_{j} − x_{j−1}))] (2.28)
|a_{j}(t)| ≤ C_{4}[exp(t(x_{j+2} − x_{j+1})) + exp(t(x_{j+1} − x_{j}))
+ exp(t(x_{j} − x_{j−1}))]  (2.29)

Remark. As explained above (see (2.21) and (2.22)), (2.28) is optimal, while (2.29) is not quite, although it has the proper e^{−tc} behavior.

Proof. (2.28) follows from (2.17) and (2.25), while (2.29) follows from (2.18) and (2.25). □

3. A Theorem of Killip and Nenciu

In this section, we want to prove Theorem 1.5. We will follow the strategy of the last section with some changes necessitated by the fact the K-groups can have more than one point. In particular, we cannot use mere counting to be sure only one zero approaches a single pure point. Instead we will need the following theorem of Denisov–Simon that appears as Theorem 1.7.20 of [17]:

Theorem 3.1. Let z_{0} be an isolated point of the support of a probability measure on ∂𝔻. Let

d = dist(z_{0}, supp(dμ)\{z_{0}})  (3.1)

Then each OPUC, Φ_{j}(z; dμ) has at most one zero in the circle of radius d^{2}/6 about z_{0}.

Remarks. 1. For OPRL, isolated points of the support can have two nearby zeros; see [7].

2. See Appendix B for an alternate proof (of a stronger result, namely, getting d^{2}/4 rather than d^{2}/6) that uses operator theory.

Lemma 3.2. The zeros z_{k}^{(n)} of Φ_{n}(z; dμ) for any dμ on the unit circle obey

z_{k}^{(n)} = \frac{∫ z ∏_{ℓ≠k}|z − z_{ℓ}^{(n)}|^{2} dμ(θ)}{∫ ∏_{ℓ≠k}|z − z_{ℓ}^{(n)}|^{2} dμ(θ)}  (3.2)

In particular, for any y,

|z_{k}^{(n)} − y| ≤ \frac{∫ |z − y| ∏_{ℓ≠k}|z − z_{ℓ}^{(n)}|^{2} dμ(θ)}{∫ ∏_{ℓ≠k}|z − z_{ℓ}^{(n)}|^{2} dμ(θ)}  (3.3)

Remarks. 1. If dμ is trivial with N points in its support, we need n ≤ N.

2. In the integrals, z = e^{iθ}.

3. (3.2) is (1.7.51) of [17]. Again we give a variational proof.

Proof. In ∫ ∏_{ℓ=1}^{n}|z − y_{ℓ}|^{2} dμ(θ), which is minimized by y_{ℓ} = z_{ℓ}^{(n)}, the y_{ℓ}’s are complex so we can write it as a function of y_{ℓ} and ȳ_{ℓ} and demand all ∂/∂ȳ_{ℓ} and ∂/∂y_{ℓ} vanish at y_{ℓ} = z_{ℓ}^{(n)}. (3.2) comes from the ∂/∂ȳ_{k} derivative (or conjugate of the ∂/∂y_{k} derivative). (3.3) follows immediately from (3.2). □

Theorem 3.3. Let {z_{j}^{(n)}(t)}_{j=1}^{n} be the zeros of Φ_{n}(z; dμ_{t}) where n ∈ K_{m}, μ_{t} is given by (1.13), and dμ_{t=0} has finite support. Here K_{m} are the K-group defined after (1.17). Then for t large, Φ_{n} has exactly one zero near each {z_{j}}_{j=1}^{k_{m−1}} which, by renumbering, we can suppose are z_{j}^{(n)}(t). Moreover, for 1 ≤ j ≤ k_{m−1},

|z_{j}^{(n)}(t) − z_{j}| ≤ C exp(t[G(z_{n}) − G(z_{j})])  (3.4)

Proof. The proof is identical to Theorem 2.4, given Theorem 3.1 to be sure ∏_{ℓ≠j}|z − z_{ℓ}^{(n)}|^{2} stays away from the zero at z = z_{j}. □

To continue, we will need a lovely consequence of the Szegő variational principle that will also be the key to the arguments in Section 4. Recall that one can define monic OPUC for any positive measure, even if not normalized, and, of course,

Φ_{j}(z; c dμ) = Φ_{j}(z; dμ)  (3.5)

for any positive constant c.

Theorem 3.4. Let dμ be a nontrivial measure on ∂𝔻 and let {z_{j}}_{j=1}^{k} be among the zeros of Φ_{n}(z; dμ). Then

Φ_{n}(z; dμ) = ∏_{j=1}^{k}(z − z_{j})Φ_{n−k}(z; ∏_{j=1}^{k} |z − z_{j}|^{2} dμ)  (3.6)

Remark. The z_{j}’s can be repeated up to their multiplicity.

Proof. Let Q_{n−k} be a monic polynomial of degree n − k, so ∏_{j=1}^{k}(z − z_{j})Q_{n−k} is a monic polynomial of degree n. Thus, by the Szegő variational principle,

∫ |\frac{Φ_{n}(z; dμ)}{∏_{j=1}^{k}(z − z_{j})}|^{2} ∏_{j=1}^{k} |z − z_{j}|^{2} dμ = ∫ |Φ_{n}(z; dμ)|^{2} dμ

≤ ∫ | ∏_{j=1}^{k}(z − z_{j})Q_{n−k}(z)|^{2} dμ

= ∫ |Q_{n−k}(z)|^{2} ∏_{j=1}^{k} |z − z_{j}|^{2} dμ

Since Φ_{n}(z; dμ)/ ∏_{j=1}^{k}(z − z_{j}) is a monic polynomial of degree n − k and Q is arbitrary, the Szegő variational principle implies that

\frac{Φ_{n}(z; dμ)}{∏_{j=1}^{k}(z − z_{j})} = Φ_{n−k}(z; ∏_{j=1}^{k} |z − z_{j}|^{2} dμ)  (3.7)

which is (3.6). □

Remarks. 1. One can also prove this using orthogonality. For if ℓ < n − k, z^{ℓ} ∏_{j=1}^{k}(z − z_{j}) ⊥ Φ_{n}(z; dμ), so

∫ \frac{Φ_{n}(z; dμ)}{∏_{j=1}^{k}|z − z_{j}|^{2}} \overline{z^{ℓ} ∏_{j=1}^{k}(z − z_{j})} ∏_{j=1}^{k} |z − z_{j}|^{2} dμ = 0  (3.8)

but

LHS of (3.8) = ∫ \overline{z^{ℓ}} \frac{Φ_{n}(z; dμ)}{∏_{j=1}^{k}(z − z_{j})} ∏_{j=1}^{k} |z − z_{j}|^{2} dμ

proving (3.7).

2. (3.7) for n − k = 1 is easily seen to be equivalent to (3.2).

Theorem 3.5. Under the hypotheses of Theorem 1.5 with dμ_{t} = Σ_{t}(dμ) and ℓ ∈ K_{j},

lim_{t→∞} Φ_{ℓ}(z; dμ_{t}) = ∏_{p=1}^{k_{j−1}}(z − z_{p})Φ_{ℓ−k_{j−1}−1}(z; dμ^{(K_{j})})  (3.9)

Proof. Let {z_{p}(t)}_{p=1}^{k_{j−1}} be the zeros of Φ_{ℓ}(z; dμ_{t}) which converge to {z_{p}}_{p=1}^{k_{j−1}} as t → ∞. By (3.6) and (3.5) for n ∈ K_{j},

Φ_{ℓ}(z; dμ_{t}) = ∏_{p=1}^{k_{j−1}}(z − z_{p}(t))Φ_{ℓ−k_{j−1}}(z; e^{−tG(z_{n})} ∏_{p=1}^{k_{j−1}} |z − z_{p}(t)|^{2} dμ_{t}(z))

(3.10)

On account of (3.4), the weights of {z_{p}}_{p=1}^{k_{j−1}} in the measure on the right in (3.10) are bounded by

e^{2t(G(z_{n})−G(z_{y}))}e^{−tG(z_{n})}e^{+tG(z_{j})} → 0

since G(z_{j}) < G(z_{n}). The weights of points in K_{j+1}, …, K_{ℓ} go to zero. Thus

e^{−tG(z_{n})} ∏_{p=1}^{k_{j−1}} |z − z_{p}(t)|^{2} dμ_{t} → C dμ^{(K_{j})}

where C is a constant and the convergence is weak. For we have shown the contributions of z_{ℓ} ∉ K_{j} go to zero and the weights at z_{ℓ} ∈ K_{j} converge to ∏_{p=1}^{k_{j−1}}|z_{ℓ} − z_{p}|^{2}μ({z_{ℓ}}) since G(z_{n}) = G(z_{ℓ}) and z_{p}(t) → z_{p}.

(3.9) is immediate by continuity of OPs for index less or equal to the number of points in the support of the limiting measure. □

Remark. As stated, this theorem required dμ_{t=0} (and so dμ_{t}) have finite support. However, the proof works without change if dμ_{t=0} = dν_{1} +dν_{2} with dν_{1} finite and dν_{2} such that

sup_{z∈supp(dν_{2})} G(z) < min_{z∈supp(dν_{1})} G(z)

We will need this extended version later in the paper.

Proof of Theorem 1.5. Let z = 0 in (3.9) and use (1.23). □

Killip–Nenciu [12] obtain a limiting formula that involves sums of determinants, but one can manipulate Heine’s formula (see (1.5.80) of [17]) to see they have really found α_{ℓ−k_{j−1}−1}(dμ^{(K_{j})}). In fact, earlier in their proof they essentially do an inverse of this process.

4. Asymptotics of Real Schur Flows

In this section—the main one from the point of view of (1.1)—we will prove Theorem 1.1 and Proposition 1.2. The central object will be the nontrivial probability measure dμ with

α_{n}(dμ) = α_{n}(t = 0)  (4.1)

In terms of dμ, we will be able to specify which case of Theorem 1.1 holds. Define

Θ(dμ) = min{|θ| | e^{iθ} ∈ σ_{ess}(dμ)}  (4.2)

The central role of such extreme points of σ_{ess} for Toda flows was understood by Deift–Li–Tomei [5]. Basically, if x_{N} is interpreted as lim_{n→∞} x_{n} when N = ∞, we will have

x_{N} = cos(Θ(dμ))  (4.3)

We begin by analyzing the case Θ = 0.

Theorem 4.1 (Case (i) of Theorem 1.1). Suppose 1 ∈ σ_{ess}(dμ). Then for all n and j = 1, …, n,

z_{j}^{(n)}(t) → 1  (4.4)

and

α_{n}(t) → (−1)^{n}  (4.5)

Proof. (4.5) follows from (4.4) and (1.23). Let z_{1}^{(n)}(t), …, z_{n}^{(n)}(t) be the zeros of Φ_{n}(x; dμ_{t}). By (3.6),

z − z_{j}^{(n)}(t) = Φ_{1}(z; N_{t}^{−1} ∏_{k≠j} |z − z_{k}^{(n)}(t)|^{2} dμ)  (4.6)

where N_{t} is a normalization. By Lemma 4.2 below, the measure on the right converges to δ_{z=0}, so by continuity of α_{n}(·) under weak convergence (with the critical addendum that if the limit has k pure points, it only holds for n = 0, 1, …, k − 1),

α_{0}(N_{t}^{−1} ∏_{k≠j} |z − z_{k}^{(n)}(t)|^{2} dμ) → 1

so, since

Φ_{1} = z − ᾱ_{0}

we conclude

z_{j}^{(n)}(t) = ᾱ_{0} → 1  (4.7)

Since j is arbitrary, we have proven (4.4). □

Remark. For this case, where we only need information of Φ_{1}, one can use (3.2) instead of (3.6). By (3.2),

z_{j}^{(n)} = \frac{∫ z ∏_{k≠j}|z − z_{k}^{(n)}|^{2} dμ}{∫ ∏_{k≠j}|z − z_{k}^{(n)}|^{2} dμ} → 1

by Lemma 4.2. We used (3.6) since it is needed for the later arguments.

Lemma 4.2. For any nontrivial probability measure dμ on ∂𝔻 with 1 ∈ σ_{ess}(dμ) and any w_{1}(t), …, w_{ℓ}(t) ∈ \overline{𝔻}, we have

N_{t}^{−1} ∏_{j=1}^{ℓ} |z − w_{j}(t)|^{2} dμ_{t} → δ_{z=1}  (4.8)

as t → ∞. Here N_{t} = ∫ ∏_{j=1}^{ℓ}|z − w_{j}(t)|^{2} dμ_{t}.

Proof. The idea is that in μ_{t}, points near zero have much stronger weight than fixed intervals away from zeros. The |z − w_{ℓ}(t)|^{2} factors can overcome that difference (and, as we have seen in the finite case, do if dμ has an isolated pure point at z = 1), but to do this, the w_{ℓ}(t) have to be exponentially close to the points they mask. Thus, the finite number, ℓ, of zeros can mask only an exponentially small piece of the part of dμ_{t} near z = 1, and since 1 ∈ σ_{ess}(dμ), there are always unmasked pieces.

To be explicit, let dμ̃_{t} be the measure on the left side of (4.8). If we prove for each θ_{0} ∈ (0, π),

μ̃_{t}({e^{iη} | θ_{0} < η < 2π − θ_{0}}) → 0  (4.9)

then, by compactness, (4.8) holds. Given θ_{0}, since 1 ∈ σ_{ess}(dμ), we can find

θ_{0} > θ_{1} > φ_{1} > θ_{2} > φ_{2} > ⋯ > θ_{ℓ+1} > φ_{ℓ+1} > 0  (4.10)

so that

μ({e^{iη} | θ_{j} > η > φ_{j}}) > 0  (4.11)

for each j = 1, …, ℓ + 1. Call the set in (4.11), I_{j}.

Let

Q_{j}(t) = min_{η∈I_{j}} ∏_{m=1}^{ℓ} |e^{iη} − w_{m}(t)|^{2}  (4.12)

Then, for each j,

LHS of (4.9) ≤ e^{2t(cos θ_{0}−cos θ_{j})}μ(I_{j})^{−1}Q_{j}(t)  (4.13)

The right side of (4.13) goes to zero unless Q_{j}(t) goes to zero as fast as e^{2t(cos θ_{0}−cos θ_{j})} and this can only happen if at least one w_{j}(t) is within e^{2t(cos θ_{0}−cos θ_{j})/ℓ} of I_{j}. Since there are ℓ + 1 intervals a finite distance from each other and only ℓ zeros, for all large t, at least one of the RHS of (4.13) (for j = 1, …, ℓ + 1) goes to zero, proving (4.9). □

The exact same argument proves:

Lemma 4.3. Let dμ be a nontrivial probability measure on ∂𝔻 invariant under z → z̄. Suppose Θ(dμ) = θ_{∞} > 0 and {e^{iη} | |η| < θ_{∞}} has finitely many pure points of dμ: {e^{iθ_{k}}}_{k=1}^{K} (either one θ_{k} = 0 and K is odd with ±θ_{k} terms or no θ_{k} is zero and K is even with ±θ_{k} terms).

Let ℓ ≥ K and w_{1}(t), …, w_{ℓ}(t) be a conjugation-invariant set of points in ℂ so that for j = 1, …, K,

|e^{iθ_{j}} − w_{j}(t)|^{2} ≤ Ce^{(2+ε)t[cos(Θ)−cos(θ_{j})]}  (4.14)

for some ε > 0. Then

N_{t}^{−1} ∏_{j=1}^{ℓ} |z − w_{j}(t)|^{2} dμ_{t} → \frac{1}{2}(δ_{z=e^{iΘ}} + δ_{z=e^{−iΘ}})  (4.15)

Remark. To get \frac{1}{2} on the right in (4.15), we use the fact that since dμ_{t} and {w_{j}(t)} are conjugation symmetric, the limit which lives on {e^{±iΘ}} must also be conjugation symmetric.

With this lemma, we can prove

Theorem 4.4. Let dμ be a nontrivial conjugation-symmetric probability measure on ∂𝔻 with Θ(dμ) = θ_{∞} > 0 and suppose there are only finitely many points {e^{iθ_{k}}}_{k=1}^{K} in {e^{iη} | |η| < θ_{∞}} in the support of dμ. Let α_{n}(t) solve (1.1) with

α_{n}(0) = α_{n}(dμ)  (4.16)

If K = 2m + 1 (i.e., θ_{1} = 0) and 0 < θ_{2} < ⋯ < θ_{m+1} and θ_{m+2} = −θ_{m+1}, θ_{m+3} = −θ_{m}, …, θ_{2m+1} = −θ_{2}, then (1.5) holds with N = m+1 and

x_{j} = cos(θ_{j+1})  j = 1, …, m  (4.17)
x_{N} = cos(θ_{∞})  (4.18)

If K = 2m and 0 < θ_{1} < ⋯ < θ_{m}, and θ_{m+1} = −θ_{m}, …, θ_{2m} = −θ_{1}, then (1.6) holds with N = m + 1 and

x_{j} = cos(θ_{j})  j = 1, …, m  (4.19)

and x_{N} given by (4.18).

Proof. We will prove the case K = 2m + 1. The other case is essentially identical. We essentially have a one-element K_{1}-group {z = 1} and m two-element K-groups, K_{2}, …, K_{m+1}, with K_{j} = {e^{±iθ_{j}}}. The analysis of Section 3 (see the remark following Theorem 3.5) works for {α_{n}(t)}_{n=0}^{2m} and proves that for any ℓ > 2m + 1, Φ_{ℓ}(z; dμ) has zeros exponentially close to {e^{iθ_{k}}}_{k=1}^{K} in the sense of (4.14) (indeed, one can take 2 + ε = 4).

Thus for any ℓ ≥ 0,

N_{t}^{−1} ∏_{j=1}^{2m+1} |z − z_{j}^{(2m+1+ℓ)}|^{2} dμ_{t} → \frac{1}{2} (δ_{z=e^{iΘ}} + δ_{z=e^{−iΘ}}) ≡ dη (4.20)

so by (3.6) for ℓ = 1, 2,

Φ_{2m+1+ℓ}(z) → ∏_{j=1}^{2m+1}(z − z_{j})Φ_{ℓ}(z; dη)  (4.21)

The two-point measure has α_{0} = cos(Θ) and α_{1} = −1, which proves the formula for α_{2m+1}(dμ_{t}) and α_{2m+2}(dμ_{t}).

By Lemma 4.5 below and the argument in the first paragraph, we know that for any ℓ ≥ 0, that 2m + 3 zeros of Φ_{2m+3+ℓ}(t) approach {e^{iθ_{k}}}_{k=1}^{K} ∪ {e^{±iΘ}}. Repeating the argument above, we get α_{2m+3}(dμ_{t}) and α_{2m+4}(dμ_{t}). Iterating, we get α_{2m+ℓ}(dμ) for all ℓ. □

Lemma 4.5. If dμ_{t} is a family of measures indexed by t ∈ (0, ∞) and for some fixed N, there are {z_{j}^{(∞)}}_{j=1}^{N} ∈ ∂𝔻 so that the zeros {z_{j}^{(n)}(t)}_{j=1}^{N} of Φ_{N}(z; dμ_{t}) approach {z_{j}^{(∞)}}_{j=1}^{N}, then for any ℓ > N, there are N zeros of Φ_{ℓ}(z; dμ_{t}) which approach {z_{j}^{(∞)}}_{j=1}^{N}.

Proof. Since the coefficients of Φ_{j}(z; dν) are uniformly bounded by 2^{j} (uniformly in dν by Szegő recursion), Φ_{ℓ}(z; dμ) are uniformly bounded analytic functions. So it suffices to show for each j, Φ_{ℓ}(z_{j}^{(∞)}; dμ_{t}) → 0.

Since Φ_{N}(z; dμ_{t}) → ∏_{j=1}^{N}(z − z_{j}^{(∞)}), we have

Φ_{N}^{∗}(z; dμ_{t}) → ∏_{j=1}^{N}(1 − zz̄_{j}^{(∞)}) = ∏_{j=1}^{N}(−z̄_{j}^{(∞)})Φ_{N}(z; dμ_{t})  (4.22)

Thus

Φ_{N}^{∗}(z_{j}^{(∞)}) → 0 Φ_{N}(z_{j}^{(∞)}) → 0  (4.23)

By Szegő recursion,

Φ_{N+1}^{∗}(z_{j}^{(∞)}) → 0 Φ_{N+1}(z_{j}^{(∞)}) → 0  (4.24)

so by induction,

Φ_{N+m}^{∗}(z_{j}^{(∞)}) → 0 Φ_{N+m}(z_{j}^{(∞)}) → 0  (4.25)

for all m. □

We summarize in a strong version of Theorem 1.1:

Theorem 4.6. Suppose each α_{n}(0) ∈ (−1, 1) and let dμ be the measure with α(dμ) = α_{n}(0) (which is conjugation-symmetric). Then

(i) If 1 ∈ σ_{ess}(dμ), (1.3) holds. (ii) If Θ(dμ) > 0, 1 ∉ supp(dμ), and {e^{iη} | |η| < Θ} has 2m points,

then (1.6) holds with N = m + 1 and x_{j} is given by (4.19) and (4.18).

(iii) If Θ(dμ) > 0, 1 ∉ supp(dμ), and {e^{iη} | |η| < Θ} has an infinity of points, then (1.6) holds with N = ∞ and x_{j} is given by (4.18) and x_{j} → cos(Θ) as N → ∞.

(iv) If Θ(dμ) > 0, 1 ∈ supp(dμ), and {e^{iη} | |η| < Θ} has 2m+1 points,

then (1.5) holds, N = m + 1, and θ_{j} is given by (4.17) and (4.18).

(v) If Θ(dμ) > 0, 1 ∈ supp(dμ), and {e^{iη} | |η| < Θ} has an infinity of points, then (1.5) holds with N = ∞ and x_{j} is given by (4.17) and x_{j} → cos(Θ) as N → ∞.

Proof. (i)–(iii) are proven in Theorems 4.1 and 4.4. (iv)–(v) follow from the method of Section 3 with no need for analysis of the edge of the essential spectrum. □

Proof of Proposition 1.2. (i) By Theorem 4.3.17 of [17], α_{n}(0) → 0 implies supp(dμ) = ∂𝔻 implies 1 ∈ σ_{ess}(dμ).

(ii) By Theorem 4.2.11 of [17], if α_{n+1}ᾱ_{n} → −1, σ_{ess}(dμ) = {1}, so 1 ∈ σ_{ess}(dμ).

(iii),(iv) By Example 4.3.10 of [17], σ_{ess}(dμ) = [θ_{0}, 2π − θ_{0}] where cos θ_{0} = 1 − 2a^{2}. So the only issue is whether 1 ∈ supp(dμ) or not. Since α_{n}(0) ∼ ∏_{j=0}^{n}(1−α_{j}) (for α_{j} real), we see that α_{j} → a > 0 means 1 ∈ supp(dμ) and α_{j} → a < 0 means 1 ∉ supp(dμ).

(v),(vi) By Theorem 4.2.11 of [17], σ_{ess}(dμ) = {−1} if and only if α_{n+1}ᾱ_{n} → 1, which for α_{n} real means α_{n} → 1 or α_{n} → −1. If α_{n} → 1 (resp. −1), 1 ∈ supp(dμ) (resp. 1 ∉ supp(dμ)) by the argument used for (iii) and (iv). □

5. Higher-Order Asymptotics in Case (i)

For the special case α_{n}(t = 0) ≡ 0, Golinskii proved that

(−1)^{n}α_{n}(t) = 1 − \frac{n + 1}{4t} + o(\frac{1}{t})  (5.1)

(by our method below or his method, one can see o(1/t) is O(1/t^{2})). He has two proofs: one uses the difference equation (in n) obeyed by α_{n}(t) in this special case, and the other, some explicit formulae in terms of Bessel functions. Here is another proof which does not depend on special features of α_{n}(0) ≡ 0 but only depends on the form dμ near θ = 0:

Proposition 5.1. If α_{n}(0) is real and

∑_{n=0}^{∞} |α_{n}(t = 0)| < ∞  (5.2)

then (5.1) holds.

Remark. All the proof requires is that dμ = w(θ)\frac{dθ}{2π} + dμ_{s} where 0 ∉ supp(dμ_{s}) and θ = 1 is a Lebesgue point of w with positive density. (5.2) is only used to prove that.

Proof. We use the following formula (see (1.5.80) and (1.5.88) of [17]):

(−1)^{n}α_{n}(dμ) = N^{−1} ∫ e^{−i(θ_{0}+···+θ_{n})} ∏_{0≤j<k≤n}|e^{iθ_{j}}−e^{iθ_{k}}|^{2} ∏_{j=0}^{n} dμ(θ_{j}) (5.3)

where

N = ∫ ∏_{0≤j<k≤n}|e^{iθ_{j}} − e^{iθ_{k}}|^{2} ∏_{j=0}^{n} dμ(θ_{j})

By Baxter’s theorem, (see Theorem 5.2.1 of [17]), (5.2) implies

dμ_{t=0}(θ) = f(θ) \frac{dθ}{2π}  (5.4)

where f is continuous and nonvanishing.

By (5.3),

\begin{aligned}1 & − (−1)^{n}α_{n}(dμ_{t}) \\ & = N_{t}^{−1} ∫ (−e^{−i(θ_{0}+···+θ_{n})} + 1) ∏_{0≤j<k<n} |e^{iθ_{j}} − e^{iθ_{k}}|^{2} ∏_{j=0}^{n} e^{2t cos θ_{j}}f(θ_{j}) \frac{dθ_{j}}{2π} \end{aligned} ; (5.5)

For each fixed ε, we can break the integral up into the region |θ_{j}| < ε for all ε, and its complement. The integral over the complement is bounded in absolute value by Ce^{2t cos(ε)}.

Consider next the integral over the region |θ_{j}| < ε. By the θ_{j} → −θ_{j} symmetry, we can replace 1 − e^{−i(θ_{0}+···+θ_{n})} by 1 − cos(θ_{0} + ⋯ + θ_{n}) and so get a positive integrand. Picking the contribution of the region

\frac{2jε}{(n + 1)^{2}} < θ_{j} < \frac{(2j + 2)ε}{(n + 1)^{2}} ≡ η_{j}(ε)  (5.6)

we get a lower bound of the form

Cε^{2}(ε^{2})^{n(n−1)/2}e^{2t(∑_{j=0}^{n} cos η_{j})}(ε)^{n}

In this way, we see that for ε small, the ratio of the complement to the remainder is

O(ε^{−p}e^{−tDε^{2}})

for some p, D > 0 (which are n dependent). This goes to zero as t → ∞ for any fixed n, ε, so in N_{t} and the integral, we can restrict integrals to |θ_{j}| < ε and make an arbitrarily small fractional error.

Once |θ_{j}| < ε, we can replace f(θ) by f(0), e^{2t cos θ_{j}} by e^{2t−tθ_{j}^{2}}, and e^{iθ_{j}} − e^{iθ_{k}} by θ_{j} − θ_{k} with fractional errors going to zero. We conclude that

\begin{matrix}1− & (−1)α_{n}(dμ_{t}) = N_{t}^{−1} ∫ \frac{1}{2} (θ_{0} + ⋯ + θ_{n})^{2} \\ ∏_{0≤j<k≤n}|θ_{j} − θ_{k}|^{2}e^{−t(θ_{0}^{2}+···+θ_{n}^{2})} \frac{dθ_{0}}{2π} ⋯ \frac{dθ_{n}}{2π} (1 + o(1)) \end{matrix}  (5.7)

where N_{t} is the integral without the \frac{1}{2}(θ_{0} + ⋯ + θ_{n})^{2}.

To see this, we use positivity. Thus f(0) > 0 and f continuous means δ(ε) = sup_{|θ|<ε}|\frac{f(θ)}{f(0)} − 1| → 0 as ε ↓ 0, so using

f(0)(1 − δ) ≤ f(θ) ≤ f(0)(1 + δ)

in all places, we establish the claim about fractional errors in replacing f(θ) by f(0). Similarly,

lim_{ε↓0} sup_{\begin{matrix}|θ_{j}|<ε \\ |θ_{k}|<ε \end{matrix}} |\frac{e^{iθ_{j}} − e^{iθ_{k}}}{θ_{j} − θ_{k}} − 1| = 0

allowing the other fractional replacement. Because of t-dependence, the replacement is more subtle in e^{2t cos(θ_{j})}. We show there is δ(ε) so |δ| < ε means

e^{−(1+δ(ε))θ^{2}t} ≤ e^{2t(1−cos θ)} ≤ e^{−(1−δ(ε))θ^{2}t}

and that allows, after calculations below, to show, as δ ↓ 0, the replacement is allowed.

In the integrals in (5.7), we can now take θ_{j} to run from −∞ to ∞ for, by the same arguments as above, those integrals are dominated by the region |θ_{j}| < ε.

Now change variables from θ_{0}, …, θ_{n} to

x_{0} = θ_{0} + ⋯ + θ_{n+1}
x_{j} = θ_{j} − θ_{0} j = 1, …, n

Since e_{0} = δ_{0} + ⋯ + δ_{n} is a vector of Euclidean length \sqrt{n + 1} and δ_{j} − δ_{0} ⊥ e_{0}, we see

θ_{0}^{2} + ⋯ + θ_{n}^{2} = \frac{x_{0}^{2}}{n + 1} + Q(x_{1}, …, x_{n})

where Q is a positive quadratic form.

The integrands in (5.7) factor into functions of x_{0} or (x_{1}, …, x_{n}) with identical integrands in (x_{1}, …, x_{n}). Thus, those factors cancel and we are left with x_{0} integrals only, and we get:

Integral on RHS of (5.7) = \frac{\frac{1}{2} ∫ y^{2}e^{−ty^{2}/n+1} dy}{N}

= \frac{(\frac{n+1}{2}) ∫ w^{2}e^{−tw^{2}} dw}{N}

= −\frac{n + 1}{2} \frac{d}{dt}[log ∫ e^{−tw^{2}} dw]

= \frac{n + 1}{4} \frac{1}{t}

proving (5.1). □

Once we drop (5.2), the higher-order asymptotics are not universal. For example, if dμ = \frac{dθ}{4π} + \frac{1}{2}δ_{θ=0} (for which α_{n}(0) = \frac{1}{n+2}; see Example 1.6.3 of [17]), then 1 − α_{0}(t) = ct^{−3/2}. One can similarly get lots of variation in asymptotics of α_{n}(t) staying within ∑_{n=0}^{∞}|α_{n}(0)|^{2} < ∞. In fact, one can arrange that log(1 − α_{·}(t))/ log t does not have a limit as t → ∞.

6. Pathologies for Complex Initial Conditions

Our goal in this section is to provide the example that proves Proposition 1.3. As a warmup, we consider a measure on [0, ∞):

Example 6.1. Let

x_{k} = \frac{1}{k!}  (6.1)

and

ρ_{k} = e^{−k^{3/2}}  (6.2)

and

dρ_{t} = \frac{∑_{k=1}^{∞} e^{−tx_{k}}ρ_{k}δ_{x_{k}}}{∑_{k=1}^{∞} e^{−tx_{k}}ρ_{k}}  (6.3)

Finally, let

t_{n} = n!  (6.4)

Then, we claim that

∥dρ_{t_{n}} − δ_{x_{n}}∥ = O(e^{−n^{1/2}})  (6.5)

so ρ_{t=n!} is concentrated more and more at single but variable points. It is not hard to see that (6.5) is equivalent to

\frac{∑_{m≠n} e^{−t_{n}x_{m}}ρ_{m}}{e^{−t_{n}x_{n}}ρ_{n}} = O(e^{−n^{1/2}})  (6.6)

or, since t_{n}x_{n} = 1, to

∑_{m≠n} e^{−t_{n}x_{m}}ρ_{m}ρ_{n}^{−1} = O(e^{−n^{1/2}})  (6.7)

Note first that

e^{−t_{n}x_{n−j}} = e^{−n!/(n−j)!}

= \begin{cases}e^{−n} & j = 1 \\ O(e^{−n(n−1)}) & j ≥ 2 \end{cases}

where the O(…) term is uniform in j and n. On the other hand,

ρ_{n−1}ρ_{n}^{−1} = e^{n^{3/2}−(n−1)^{3/2}} = e^{O(n^{1/2})}

while

ρ_{n−j}ρ_{n}^{−1} ≤ ρ_{n}^{−1} = e^{n^{3/2}}

for j ≥ 2. It follows that

∑_{m<n} e^{−t_{n}x_{m}}ρ_{m}ρ_{n}^{−1} = e^{−n+O(n^{1/2})} + ne^{−n^{2}+O(n^{3/2})}

is certainly O(e^{−n^{1/2}}).

For m > n, we only need e^{−t_{n}x_{n+j}} ≤ 1 since

ρ_{n+j}ρ_{n}^{−1} = e^{−(n+j)^{3/2}}e^{n^{3/2}} = exp(− ∫_{n}^{n+j} \frac{3}{2} x^{1/2} dx)
= O(e^{−jn^{1/2}})

so

∑_{m>n} e^{−t_{n}x_{m}}ρ_{m}ρ_{n}^{−1} = O( ∑_{j=1}^{∞} e^{−jn^{1/2}}) = O(e^{−n^{1/2}})

This proves (6.7), and so (6.5). □

Example 6.2. We will re-use Example 6.1 but with the wrinkle that for odd n, we will put the points near i, and for even n, near −i. Explicitly, define z_{n} by

2 Re z_{n} = −\frac{1}{n!}  (6.8)
|z_{n}| = 1  (6.9)
(−1)^{n+1} Im z_{n} > 0  (6.10)

With

μ_{n} = \frac{e^{−n^{3/2}}}{∑_{j=1}^{∞} e^{−j^{3/2}}}

define

dμ = ∑_{n=1}^{∞} μ_{n}δ_{z_{n}}  (6.11)

and dμ_{t} by (1.11).

As in Example 6.1,

dμ_{t=n!} − δ_{z_{n}} = O(e^{−n^{1/2}})  (6.12)

so since

\overline{α_{0}(t)} = ∫ z dμ_{t}  (6.13)

we have

α_{0}(n!) − z̄_{n} → 0  (6.14)

But z_{2n} → −i, while z_{2n+1} → i, so

lim_{n→∞} α_{0}((2n)!) = i  lim_{n→∞} α_{0}((2n + 1)!) = −i

and lim α_{0}(t) does not exist. This proves Proposition 1.3. □

Since after z_{n}, the dominant weight at time t_{n} is at z_{n+1}, it should be possible to show, using the methods of Sections 3 and 4, that α_{1}(n!) − z̄_{n+1} → 0 and α_{j}(n!) − z̄_{n+j} → 0 so that no α_{j}(t) has a limit.

7. Schur Flows with Complex but Regular Initial Conditions

If one allows complex initial conditions, there are two cases where the pathology of the previous section does not occur. As earlier, Θ(dμ) is defined by (4.2).

Theorem 7.1. If

S = {θ | |θ| < Θ, e^{iθ} ∈ supp(dμ)}  (7.1)

is infinite, the asymptotic dynamics is determined as follows. Number the points in {θ_{j}}_{j=0}^{∞} in S so

|θ_{0}| ≤ |θ_{1}| ≤ |θ_{2}| ≤ …  (7.2)

We have that

(i) If |θ_{j−1}| < |θ_{j}| < |θ_{j+1}|, then

α_{j}(t) → (−1) ∏_{k=0}^{j}(−e^{−iθ_{k}})  (7.3)

(ii) If |θ_{j}| = |θ_{j+1}|, then

α_{j}(t) → [ ∏_{k=0}^{j−1}(−e^{−iθ_{k}})][ae^{−iθ_{j}} + (1 − a)e^{iθ_{j}}]  (7.4)

where

a = \frac{β_{+}}{β_{+} + β_{−}}  (7.5)

with

β_{±} = ∏_{k=0}^{j−1} |e^{±iθ_{j}} − e^{iθ_{k}}|^{2}μ({e^{±iθ_{j}}})  (7.6)

Theorem 7.2. If the set S of (7.1) is finite and there is a unique point, e^{iθ_{∞}}, in supp(μ) with |θ_{∞}| = Θ, then if the points {θ_{j}}_{j=0}^{N} in S are labeled so (7.2) holds, then (i), (ii) above hold for j ≤ N, and for j ≥ N + 1,

α_{j}(t) → (−1)[ ∏_{k=0}^{N}(−e^{−iθ_{k}})](−e^{−iθ_{∞}})^{j−N}  (7.7)

The proofs of these theorems are a simple modification of the arguments in Sections 3 and 4. Essentially, so long as |θ_{j}| < |θ_{j+1}|, the zeros of Φ_{j}(z; dμ_{j}) approach {e^{iθ_{k}}}_{k=0}^{j}, and in the case of a nondegenerate bottom of the essential spectrum, all the extra zeros go to e^{iθ_{∞}}.

We want to consider a case where the bottom of the essential spectrum is degenerate, that is, e^{±iθ} ∈ σ_{ess}(dμ), but μ is regular near ±Θ in a sense we will make precise. For simplicity, we will suppose the set S is empty and e^{±iΘ} are not pure points. It is easy to handle S finite or {e^{±iΘ}} eigenvalues, but it complicates the statements (and, of course, S infinite is already in Theorem 7.1). Definition. We say μ is weakly regular if there exists a_{±} ∈ (−1, ∞) so that

(i) μ({e^{iθ} | |θ| ≤ Θ}) = 0  (7.8)
(ii) lim_{ε↓0} \frac{log(μ\{e^{±iθ} | Θ ≤ θ ≤ Θ + ε\})}{log ε} = a_{±}  (7.9)

If (ii) is replaced by

(iii) lim_{ε↓0} μ({e^{iθ} | Θ ≤ θ ≤ Θ + ε})ε^{−a_{±}} = C_{±} ∈ (0, ∞)

we say μ is strongly regular.

Here are the theorems in this case:

Theorem 7.3. If μ is weakly regular with a_{+} − a_{−} ∉ 2ℤ, then the asymptotic dynamics is as follows. Suppose n ≥ 0 is an integer so a_{−} ∈ (a_{+} + 2n, a_{+} + 2n + 2) (otherwise, if a_{−} < a_{+}, interchange them):

(i) α_{j}(t) → −(−e^{−iΘ})^{j}  j = 0, 1, …, n
(ii) α_{n+2k}(t) → −(−e^{−iΘ})^{n}  k = 0, 1, …
(iii) α_{2n+2k+1}(t) → −(−e^{−iΘ})^{n−1}  k = 0, 1, …

Theorem 7.4. If μ is strongly regular and a_{+} − a_{−} ∈ 2ℤ, say a_{−} = a_{+} + 2n with n ≥ 0, then

(i) α_{j}(t) → −(−e^{−iΘ})^{j}  j = 0, …, n − 1
(ii) α_{2n+2k−1}(t) → −(−e^{−iΘ})^{n−1}  k = 0, 1, …
(iii) α_{n+2k}(t) → (−e^{−iΘ})^{n−1}(ae^{−iΘ} + (1 − a)e^{iΘ})

with a given by (7.5) with

β_{+} = C_{+}  β_{−} = |e^{iΘ} − e^{−iΘ}|^{2n−2}C_{−}  (7.10)

The proofs here are simple modifications of the arguments in Section 4. In case a_{+} − a_{−} ∉ 2ℤ, μ_{t} approaches δ_{e^{iΘ}}, then |z − e^{iΘ}|^{2}μ_{t}/N_{t}^{(1)} has δ_{e^{iΘ}} as its limit if δ_{e^{iΘ}} if a_{−} > a_{+} + 2 or δ_{e^{−iΘ}} if a_{−} < a_{+} + 2. We repeat this n times, and after that the limits alternate between δ_{e^{iΘ}} and δ_{e^{−iΘ}}. If a_{+} − a_{−} ∈ 2ℤ, we get measure modification by products of |z − z_{j}|^{2} which are aδ_{e^{iΘ}} + (1 − a)δ_{e^{−iΘ}}.

Appendix A: The Symes–Deift–Li–Tomei Representation of Dynamics

In this appendix, we will find an operator formulation of the flow Σ_{t}^{G} of (1.13). This formulation for the Toda flow was discovered by Symes [20] and then generalized to Jacobi analogs of Σ_{t}^{G} by Deift–Li–Tomei [5]. Killip–Nenciu [12] discussed this for polynomial G and finite CMV matrices. We include this appendix for four reasons. First, we wish to show that one can handle infinite CMV matrices. Second, while we regard this as a central result in symplectic flows, it is somewhat hidden in [12]. Third, the elementary identification of the spectral measure below, while implicit in the earlier works, is not made explicit. Finally, we want to note some aspects of the equivalence result (his Theorem 1) of Golinskii [11] without extensive calculation.

The QR algorithm is critical to this appendix. We will consider bounded operators on H ≡ ℓ^{2}({0, 1, 2, … }) which are therefore given by semi-infinite matrices. We use {δ_{n}}_{n=0}^{∞} for the canonical basis of H.

A bounded operator, B, is called positive upper triangular if and only if B_{rs} = 0 for r > s and B_{rr} > 0. The set of all such operators will be denoted R. The following is well known. We sketch the proof to emphasize the final formulae:

Lemma A.1. Let A be an invertible bounded operator on H. Then

A = QR  (A.1)

with Q unitary and R ∈ R. This decomposition is unique. Moreover,

Qδ_{0} = \frac{Aδ_{0}}{∥Aδ_{0}∥}  (A.2)

Proof. Uniqueness is immediate since Q_{1}R_{1} = Q_{2}R_{2} implies Q_{2}^{−1}Q_{1} = R_{2}R_{1}^{−1} and B ∈ R and unitary implies B = 1.

Let e_{0}, e_{1}, … be the set obtained by applying Gram–Schmidt to Aδ_{1}, Aδ_{2}, . . .. Note that

e_{0} = \frac{Aδ_{0}}{∥Aδ_{0}∥}  (A.3)

Because A is invertible, {e_{n}}_{n=0}^{∞} is a basis and

Aδ_{j} = ∑_{k=0}^{n} r_{kj}e_{j}  (A.4)

and

r_{kk} > 0  (A.5)

by the Gram–Schmidt construction.

Let Q be defined by

Qδ_{j} = e_{j}  (A.6)

so (A.3) becomes (A.2). By (A.4) and (A.5),

Q^{−1}A ≡ R  (A.7)

lies in R, and clearly, A = QR. □

We freely use the CMV matrix, C, and alternate CMV matrix, C̃, discussed in [17] and [19]. Here is the main result of this appendix:

Theorem A.2. Let C be a CMV matrix associated to the measure dμ. Let G be a real-valued function in L^{∞}(∂𝔻, dμ). Define Q_{t}, R_{t} by

exp(\frac{1}{2} tG(C)) = Q_{t}R_{t}  (A.8)

using the QR algorithm (A.1). Define C_{t} by

C_{t} = Q_{t}^{−1}CQ_{t}  (A.9)

Then C_{t} is the CMV matrix of the measure dμ_{t} given by (1.13).

Our proof is related to that of Killip–Nenciu [12], which in turn is a CMV analog of the results of Deift–Li–Tomei [5] for Jacobi matrices. For notational simplicity, we will deal with nontrivial dμ. [12] handles the case where dμ has finite support. We need: Definition. A matrix, M, on H is said to have CMV shape if and only if

(i) M is five-diagonal, that is, M_{jk} = 0 if |j − k| > 2 (ii) M_{2n,2n+2} > 0, n = 0, 1, 2, … (iii) M_{2n+1,2n+3} = 0, n = 0, 1, 2, … (iv) M_{2n+3,2n+1} > 0, n = 0, 1, 2, …

(v) M_{2n+2,2n} = 0, n = 0, 1, 2, … (vi) M_{10} > 0 We say M has alternate CMV shape if M^{t} has CMV shape.

For finite matrices, the following is a result of [12]:

Proposition A.3. A unitary matrix, M, has CMV shape if and only if for some sequence of Verblunsky coefficients {α_{n}}_{n=0}^{∞} ⊂ 𝔻^{∞}, M = C({α_{n}}_{n=0}^{∞}). It has alternate CMV shape if and only if M = C̃({α_{n}}_{n=0}^{∞}).

Remark. Our proof differs from [12] in that they use a Householder algorithm and we use the simpler AMR factorization (see [19]).

Proof. That a CMV matrix has CMV shape follows from the form of C; see (4.2.14) of [17].

For the converse, define α_{0} ∈ 𝔻 and ρ_{0} ∈ (0, 1) by Mδ_{0} = \binom{ᾱ_{0}}{ρ_{0}}. By (v), ρ_{0} > 0 and, by unitarity, |α_{0}|^{2} + ρ_{0}^{2} = 1. Let

Θ(α) = (\begin{matrix}ᾱ & ρ \\ ρ & −α \end{matrix})  (A.10)

and consider

(Θ(α_{0}) ⊕ 1)^{−1}M  (A.11)

It is clearly unitary and has 1 in the 11 corner, and by ρ_{0} > 0 and the definition of CMV shape, it is of the form 1_{1×1} ⊕ M_{1} where M_{1} is of alternate CMV shape. Thus

M = (Θ(α_{0}) ⊕ 1)(1_{1×1} ⊕ M_{1})  (A.12)

Applying this to get M_{1}^{t}, we see

M = (Θ(α_{0}) ⊕ 1)(1_{2×2} ⊕ M_{2})(1_{1×1} ⊕ Θ(α_{1}) ⊕ 1)  (A.13)

where M_{2} is of CMV shape. Iterating this n times,

\begin{aligned}M & = (Θ(α_{0}) ⊕ ⋯ ⊕ Θ(α_{2n−2}) ⊕ 1)(1_{2n×2n} ⊕ M_{2n}) \\ & (1_{1×1} ⊕ ⋯ ⊕ Θ(α_{1}) ⊕ Θ(α_{2}) ⊕ ⋯ ⊕ Θ(α_{2n−1}) ⊕ 1) \end{aligned} (A.14)

where M_{2n} is of CMV shape. Taking n → ∞ and taking strong limits, we see that M = C({α_{n}}_{n=0}^{∞}). □

Proof of Theorem A.2. We can write

C_{t} = R_{t}(Q_{t}R_{t})^{−1}CQ_{t}R_{t}R_{t}^{−1}
= R_{t}CR_{t}^{−1}  (A.15)

since e^{tG(C)/2} commutes with C. Since R_{t} and R_{t}^{−1} are in R, it is easy to see that (C_{t})_{jk} = 0 if k > j + 2 and the conditions (ii) and (iii) of the definition of CMV shape hold.

On the other hand, since e^{tG/2} is selfadjoint and Q unitary:

e^{tG(C)/2} = (e^{tG(C)/2})^{∗} = R_{t}^{∗}Q_{t}^{∗} = R_{t}^{∗}Q_{t}^{−1}  (A.16)

so

C_{t} = (R_{t}^{∗})^{−1}(R_{t}^{∗}Q_{t}^{−1})C(R_{t}^{∗}Q_{t}^{−1})^{−1}R_{t}^{∗}
= (R_{t}^{∗})^{−1}CR_{t}^{∗}  (A.17)

so, since R_{t}^{∗} is lower triangular and positive on diagonal, (C_{t})_{jk} = 0 if k < j − 2 and conditions (ii), (v), and (vi) hold.

Thus, C_{t} has CMV shape, and so is a CMV matrix by Proposition A.3.

The spectral measure of C_{t} and the vector δ_{0} is that of C and Q_{t}δ_{0} which, by (A.3), is that of C and e^{tG(C)/2}δ_{0}/∥e^{tG(C)/2}δ_{0}∥, which is e^{tG(θ)}dμ(θ)/ ∫ e^{tG(θ)}dμ(θ). □

Finally, when G is a Laurent polynomial, we want to discuss the associated difference equation and the associated Lax form and, in particular, show that (1.11) solves (1.1). We will not be explicit about uniqueness, but it is not hard to prove (1.1) has a unique solution. First, following the Deift–Li–Tomei [5] calculation for the Toda analog:

Proposition A.4. Define π on selfadjoint matrices on H by

π(A)_{jk} = \begin{cases}A_{jk} & j < k \\ −A_{jk} & j > k \\ 0 & j = k \end{cases}  (A.18)

so π(A) is skew-adjoint. Then the C_{t} of (A.9) is strongly C^{1} and obeys

C_{t} = [B_{t}, C_{t}]  (A.19)

where

B_{t} = π(\frac{1}{2}G(C_{t}))  (A.20)

Example A.5. For (1.11), G(θ) = 2 cos θ, G(C_{t}) = C_{t} + C_{t}^{−1} (under C ↔ e^{iθ}) and

B_{t} = \frac{1}{2} [(C_{t} + C_{t}^{−1})_{+} − (C_{t} + C_{t}^{−1})_{−}]  (A.21)

with (·)_{+} the part of the (·) above the diagonal and (·)_{−} below. (A.19)– (A.20) is, in this case, (1.21)–(1.22) of [11]. □

Proof. Since Gram–Schmidt is an algebraic operation, on {δ_{n}}_{n=0}^{∞}, Q_{t} is strongly C^{1}, and so R_{t} = Q_{t}^{∗} exp(\frac{1}{2}tG(C)) is strongly C^{1}. Clearly, Q_{t}R_{t} = e^{tG(C)/2} implies

Q̇_{t}R_{t} + Q_{t}Ṙ_{t} = \frac{1}{2} G(C)Q_{t}R_{t}  (A.22)

or

Q_{t}^{−1}Q̇_{t} + Ṙ_{t}R_{t}^{−1} = \frac{1}{2} Q_{t}^{−1}G(C)Q_{t} = \frac{1}{2} G(C_{t})  (A.23)

Since R_{t} ∈ R, Ṙ_{t}R_{t}^{−1} is upper triangular and real on the diagonal. Since Q_{t} is unitary, Q_{t}^{−1}Q̇_{t} is skew-Hermitian. It vanishes on diagonal since both Ṙ_{t}R_{t}^{−1} and \frac{1}{2}G(C_{t}) are real there and skew-Hermitian matrices are pure imaginary on diagonals. Since Ṙ_{t}R_{t}^{−1} is upper triangular,

[Q_{t}^{−1}Q̇_{t}]_{jk} = [\frac{1}{2} G(C_{t})]_{jk}  (A.24)

for j > k. Since Q_{t}^{−1}Q̇_{t} is skew-Hermitian, vanishes on diagonal, and G(C_{t}) is Hermitian, we obtain

Q_{t}^{−1}Q̇_{t} = −B_{t}  (A.25)

where B_{t} is given by (A.20).

Differentiating (A.9),

Ċ_{t} = Q_{t}^{−1}[−Q̇_{t}]Q_{t}^{−1}CQ_{t} + Q_{t}^{−1}CQ̇_{t}  (A.26)

Inserting Q_{t}Q_{t}^{−1} before the final Q̇_{t}, we see

Ċ_{t} = B_{t}C_{t} − C_{t}B_{t}

which is (A.19). □

Finally, we note the following that can be obtained by taking limits of [12] and is discussed in [9, 4]. Let H be a real Laurent polynomial, that is, ∑_{k=−n}^{n} c_{k}e^{ikθ} where c_{−k} = c̄_{k} and let

G(θ) = \frac{dH(θ)}{dθ}  (A.27)

which is also a real Laurent polynomial. Define

t_{H}({α_{j}}_{j=0}^{∞} = “Tr”(H(C(α)))  (A.28)
≡ ∑_{k=−n}^{n} c_{k}“Tr”(C^{k})  (A.29)

where “Tr” is a formal sum. While t_{H} is a formal infinite sum, ∂t_{H}/∂ᾱ_{j} is well defined since only finitely many terms depend on ᾱ_{j}. Here is what is proven in [12, 9, 4]:

Proposition A.6. Let H be a Laurent polynomial and G given by (A.27). Then dμ_{t} given by (1.11) solves

α̇_{j} = iρ_{j}^{2} \frac{∂}{∂ᾱ_{j}} (t_{H}(α))  (A.30)

Example A.7. If H(e^{iθ}) = 2 sin θ so G(e^{iθ}) = 2 cos θ, then

t_{H}(α) = “Tr”(\frac{1}{2} (C − C^{∗}))

= i^{−1}( ∑_{j=0}^{∞} [−ᾱ_{j}α_{j−1} + α_{j}ᾱ_{j−1}])

(with α_{−1} ≡ −1) and (A.30) becomes

α̇_{j} = ρ_{j}^{2}(−α_{j−1} + α_{j+1})

that is, (1.1). □

Appendix B: Zeros of OPUC near Isolated Points of the Spectrum

In this appendix, we will prove a stronger result than Theorem 3.1 using a different proof from that of Denisov–Simon presented in [17]. This will use operator theory modeled on the following operator-theoretic proof of Fejér’s theorem:

Proposition B.1 (well-known). Let A be a bounded normal operator. Then for any unit vector, φ, ⟨φ, Aφ⟩ lies in the convex hull of the support of the spectrum of A.

Proof. By the spectral theorem, φ ∈ K a subspace (the cyclic subspace generated by A, A^{∗} on φ), and there exists a probability measure μ on spec(A), and U: K → L^{2}(ℂ, dμ) unitary, so UAU^{−1} = multiplication by z, UA^{∗}U^{−1} = multiplication by z̄, and Uφ = 1. Thus

⟨φ, Aφ⟩ = ∫ z dμ(z)

clearly lies in the convex hull of spec(A). □

Theorem B.2 (Fejér’s Theorem). If dμ is a probability measure on ℂ with

∫ |z|^{n} dμ < ∞  (B.1)

for all n and Φ_{n}(z) are the monic orthogonal polynomials for dμ, then the zeros of Φ_{n} lie in the convex hull of the support of dμ.

Proof. Let P_{n} be the projection onto polynomials of degree up to n−1. Then P_{n}η = 0 for η in ranP_{n+1} if and only if η = cΦ_{n}. Thus, if Q ∈ ranP, then P_{n}[(z − z_{0})Q] if and only if cΦ_{n}(z) = (z − z_{0})Q which happens if and only if Φ_{n}(z_{0}) = 0. Thus, zeros of Φ_{n} are precisely eigenvalues of P_{n}M_{z}P_{n} ↾ ranP_{n} where M_{z} is multiplication by z.

If η is the corresponding normalized eigenvector of P_{n}M_{z}P_{n} in ranP_{n}, then z_{0} = ⟨η, P_{n}M_{z}P_{n}η⟩ = ⟨η, zη⟩ lies in the convex hull of supp(dμ) by Proposition B.1. □

We are heading towards proving the following:

Theorem B.3. Let A be a normal operator and z_{0} a simple eigenvalue so that z_{0} ∉ cvh(spec(A)\{z_{0}}) ≡ C. Here cvh is the “convex hull of.” Let z_{1} ∈ C so that |z_{0} − z_{1}| = min{|w − z_{0}| | w ∈ C}. Let P be an orthonormal projection and w_{1}, w_{2} two distinct eigenvalues of PAP ↾ ranP. Then

\frac{(w_{1} − z_{1}) · (z_{0} − z_{1})}{|z_{1} − z_{0}|^{2}} + \frac{(w_{2} − z_{1}) · (z_{0} − z_{1})}{|z_{1} − z_{0}|^{2}} ≤ 1  (B.2)

In particular,

|w_{1} − z_{0}| + |w_{2} − z_{0}| ≥ |z_{1} − z_{0}|  (B.3)

which implies there is at most one eigenvalue in

\{w | |w − z_{0}| < \frac{1}{2} |z_{1} − z_{0}|\}  (B.4)

Remark. |z_{1} − z_{0}| = dist(z_{0}, C).

Proof. Let B = PAP. Pick η_{1}, η_{2} to be normalized eigenvalues of B and B^{∗}, so

Bη_{1} = w_{1}η_{1} B^{∗}η_{2} = w̄_{2}η_{2}  (B.5)

and let φ be the normalized simple eigenvector of A with

Aφ = z_{0}φ A^{∗}φ = z̄_{0}φ  (B.6)

(since A is normal, if φ obeys Aφ = z_{0}φ, then A^{∗}φ also has A(A^{∗}φ) = z_{0}A^{∗}φ so A^{∗}φ = wφ and then ⟨φ, A^{∗}φ⟩ = ⟨Aφ, φ⟩ implies w = z̄_{0}).

By the spectral theorem for A,

w_{1} = ⟨η_{1}, Aη_{1}⟩ = z_{0}|⟨φ, η_{1}⟩|^{2} + (1 − |⟨φ, η_{1}⟩|^{2})x_{1}  (B.7)

where x_{1} ∈ C. Here, if Q is the projection onto multiples of φ_{1}, then x_{1} = ⟨(1 − Q)η_{1}, A(1 − Qη_{1}⟩/∥(1 − Q)η_{1}∥.

By (B.7),

w_{1} − z_{1} = (1 − |⟨φ, η_{1}⟩|^{2})(x_{1} − z_{1}) + |⟨φ, η_{1}⟩|^{2}(z_{0} − z_{1})  (B.8)

Since z_{1} minimizes |z_{0} − z_{1}| and C is convex, C is in the half-space orthogonal to z_{0} − z_{1} not containing z_{0} and so (x_{1} − z_{1})·(z_{0} − z_{1}) ≤ 0. Thus

(w_{1} − z_{1}) · (z_{0} − z_{1}) ≤ |z_{0} − z_{1}|^{2}|⟨φ, η_{1}⟩|^{2}  (B.9)

Similarly, using

w_{2} = \overline{⟨η_{2}, A^{∗}η_{2}⟩} = ⟨η_{2}, Aη_{2}⟩

we see

(w_{2} − z_{1})·(z_{0} − z_{1}) ≤ |z_{0} − z_{1}|^{2} |⟨φ, η_{2}⟩|^{2}  (B.10)

Thus, (B.2) is equivalent to

|⟨φ, η_{1}⟩|^{2} + |⟨φ, η_{2}⟩|^{2} ≤ 1  (B.11)

Next, note that

(w_{1} − w_{2})⟨η_{2}, η_{1}⟩ = ⟨η_{2}, w_{1}η_{1}⟩ − ⟨w̄_{2}η_{2}, η_{1}⟩
= ⟨B^{∗}η_{2}, η_{1}⟩ − ⟨η_{2}, Bη_{1}⟩ = 0

so ⟨η_{2}, η_{1}⟩ = 0 and (B.7) is Bessel’s inequality. This proves (B.2). Obviously,

\frac{(z_{0} − z_{1}) · (z_{0} − z_{1})}{|z_{0} − z_{1}|^{2}} + \frac{(z_{0} − z_{1}) · (z_{0} − z_{1})}{|z_{0} − z_{1}|^{2}} = 2  (B.12)

so subtracting (B.2) from (B.8),

\frac{(z_{0} − w_{1}) · (z_{0} − z_{1})}{|z_{1} − z_{0}|^{2}} + \frac{(z_{0} − w_{2})(z_{0} − z_{1})}{|z_{1} − z_{0}|^{2}} ≥ 1

from which (B.3) follows by the Schwartz inequality.

That the set (B.4) can contain at most one w is immediate from (B.3) since w_{1}, w_{2} ∈ the set (B.4) violates (B.3). □

We can now improve the error of Denisov–Simon (who got δ/3 where we get δ/2):

Theorem B.4. Let dμ obey (B.1) for all n and let Φ_{n} be the monic orthogonal polynomials. Suppose that z_{0} is a pure point of dμ and

δ = dist(z_{0}, cvh(supp(dμ)\({z_{0}}))) > 0

Then Φ_{n} has at most one zero in {z | |z − z_{0}| < δ/2}.

Proof. Let A be multiplication by z on L^{2}(ℂ, dμ) and P_{n} be the projection used in the proof of Theorem B.2. Then Theorem B.3 immediately implies the result. □

Specializing to OPUC:

Theorem B.5. Let dμ be a probability measure on ∂𝔻 and {Φ_{n}}_{n=0}^{N} the monic OPUC (if dμ is nontrivial, N = ∞; if dμ has finite support, N = #(supp(dμ))). Let z_{0} be an isolated point of ∂𝔻 and d = dist(z_{0}, supp(dμ)\{z_{0}}). Then for each fixed n < N, {z | |z − z_{0}| < d^{2}/4} has at most one zero of Φ_{n}(z).

Proof. As proven in [17], δ < d^{2}/2. □

Acknowledgements

It is a pleasure to thank Andrei Martínez-Finkelshtein, Irina Nenciu, Paul Nevai, and Vilmos Totik, and especially Leonid Golinskii for useful discussions and correspondence.

References

[1] M. J. Ablowitz and J. F. Ladik, Nonlinear differential-difference equations, J. Math. Phys. 15 (1975), 598–603.

[2] G. S. Ammar and W. B. Gragg, Schur flows for orthogonal Hessenberg matrices, Hamiltonian and Gradient Flows, Algorithms and Control, Fields Inst. Commun. 3, (1994), American Math. Soc. Providence, RI, pp. 27–34.

[3] J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12 (1999), 1119–1178.

[4] M. J. Cantero and B. Simon, in preparation.

[5] P. Deift, L. C. Li, and C. Tomei, Toda flows with infinitely many variables, J. Funct. Anal. 64 (1985), 358–402.

[6] P. Deift, T. Nanda, and C. Tomei, Ordinary differential equations and the symmetric eigenvalue problem, SIAM J. Numer. Anal. 20 (1983), 1–22.

[7] S. Denisov and B. Simon, Zeros of orthogonal polynomials on the real line, J. Approx. Theory 121 (2003), 357–364.

[8] L. Faybusovich and M. Gekhtman, On Schur flows, J. Phys. A 32 (1999), 4671–4680.

[9] M. Gekhtman and I. Nenciu, in preparation.

[10] Ya. L. Geronimus, Polynomials Orthogonal on a Circle and Their Applications, Amer. Math. Soc. Translation 1954 (1954), no. 104, 79 pp.

[11] L. Golinskii, Schur flows and orthogonal polynomials on the unit circle, to appear in Mat. Sb.

[12] R. Killip and I. Nenciu, CMV: The unitary analogue of Jacobi matrices, to appear in Commun. Pure Appl. Math.

[13] K. T.-R. McLaughlin and P. D. Miller, The dbar steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with fixed and exponentially varying nonanalytic weights, preprint.

[14] J. Moser, Finitely many mass points on the line under the influence of an exponential potential—an integrable system, Dynamical Systems, Theory and Applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), Lecture Notes in Phys., Vol. 38, Springer, Berlin, 1975, pp. 467–497.

[15] J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations, Advances in Math. 16 (1975), 197–220.

[16] B. Simon, OPUC on one foot, Bull. Amer. Math. Soc. 42 (2005), 431–460.

[17] B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, AMS Colloquium Series, American Mathematical Society, Providence, RI, 2005.

[18] B. Simon, Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory, AMS Colloquium Series, American Mathematical Society, Providence, RI, 2005.

[19] B. Simon, CMV matrices: Five years after, to appear in Proc. W. D. Evans 65th Birthday Conference.

[20] W. W. Symes, The QR algorithm and scattering for the finite nonperiodic Toda lattice, Phys. D 4 (1981/82), 275–280.

[21] G. Szegő, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., Vol. 23, American Mathematical Society, Providence, RI, 1939; 3rd edition, 1967.

E-mail address: bsimon@caltech.edu