THE ESSENTIAL SPECTRUM OF SCHRÖDINGER, JACOBI, AND CMV OPERATORS

YORAM LAST^{1,3} AND BARRY SIMON^{2,3}

Abstract. We provide a very general result that identifies the essential spectrum of broad classes of operators as exactly equal to the closure of the union of the spectra of suitable limits at infinity. Included is a new result on the essential spectra when potentials are asymptotic to isospectral tori. We also recover with a unified framework the HVZ theorem and Krein’s results on orthogonal polynomials with finite essential spectra.

1. Introduction

One of the most simple but also most powerful ideas in spectral theory is Weyl’s theorem, of which a typical application is (in this introduction, in order to avoid technicalities, we take potentials bounded):

Theorem 1.1. If V, W are bounded functions on ℝ^{ν} and lim_{|x|→∞}[V (x) − W (x)] = 0, then

σ_{ess}(−Δ + V) = σ_{ess}(−Δ + W)  (1.1)

Our goal in this paper is to find a generalization of this result that allows “slippage” near infinity. Typical of our results are the following:

Theorem 1.2. Let V be a bounded periodic function on (−∞, ∞) and H_{V} the operator −\frac{d^{2}}{dx^{2}} + V (x) on L^{2}(ℝ). For x > 0, define W (x) = V (x + \sqrt{x}) and let H_{W} be −\frac{d^{2}}{dx^{2}} + W (x) on L^{2}(0, ∞) with some selfadjoint boundary conditions at zero. Then

σ_{ess}(H_{W}) = σ(H_{V})  (1.2)

Date: April 12, 2005. 1 Institute of Mathematics, The Hebrew University, 91904 Jerusalem, Israel. E-

mail: ylast@math.huji.ac.il. Supported in part by THE ISRAEL SCIENCE FOUN-DATION (grant no. 188/02).

2 Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125. E-mail: bsimon@caltech.edu. Supported in part by NSF grant DMS-0140592.

3 Research supported in part by Grant No. 2002068 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.

Theorem 1.3. Let α be irrational and let H be the discrete Schrödinger operator on ℓ^{2}(ℤ) with potential λ cos(αn). Let H̃ be the discrete Schrödinger operator on ℓ^{2}(\{0, 1, 2, … \}) with potential λ cos(αn+\sqrt{n}). Then

σ_{ess}(H̃) = σ(H)  (1.3)

Our original motivation in this work was extending a theorem of Barrios-López [9] in the theory of orthogonal polynomials on the unit circle (OPUC); see [76, 77].

Theorem 1.4 (see Example 4.3.10 of [76]). Let {α_{n}}_{n=0}^{∞} be a sequence of Verblunsky coefficients so that for some a ∈ (0, 1), one has

lim_{n→∞} |α_{n}| = a  lim_{n→∞} \frac{α_{n+1}}{α_{n}} = 1  (1.4)

Then the CMV matrix for α_{n} has essential spectrum identical to the case α_{n} ≡ a.

This goes beyond Weyl’s theorem in that α_{n} may not approach a; rather |α_{n}| → a but the phase is slowly varying and may not have a limit. The way to understand this result is to realize that α_{n} ≡ a is a periodic set of Verblunsky coefficients. The set of periodic coefficients with the same essential spectrum is, for each λ ∈ ∂𝔻 (𝔻 = {z | |z| < 1}), the constant sequence α_{n} = λa. (1.5) says in a precise sense that the given α_{n} is approaching this isospectral torus. We wanted to prove, and have proven, the following:

Theorem 1.5. If a set of Verblunsky coefficients or Jacobi parameters is asymptotic to an isospectral torus, then the essential spectrum of the corresponding CMV or Jacobi matrix is identical to the common essential spectrum of the isospectral torus.

In Section 5, we will be precise about what we mean by “asymptotic to an isospectral torus.” Theorem 1.5 positively settles Conjecture 12.2.3 of [77].

In the end, we found an extremely general result. To describe it, we recall some ideas in our earlier paper [48]. We will first consider Jacobi matrices (b_{n} ∈ ℝ, a_{n} > 0)

J = (\begin{matrix}b_{1} & a_{1} & 0 & ⋯ \\ a_{1} & b_{2} & a_{2} & ⋯ \\ 0 & a_{2} & b_{3} & ... \\ ⋮ & ⋮ & ... & ... \end{matrix})  (1.5)

where, in line with our convention to deal with the simplest cases in this introduction, we suppose there is a K ∈ (0, ∞) so

sup_{n} |b_{n}| + sup_{n} |a_{n}| + sup_{n} |a_{n}|^{−1} ≤ K  (1.6)

A right limit point of J is a double-sided Jacobi matrix, J^{(r)}, with parameters {a_{n}^{(r)}, b_{n}^{(r)}}_{n=−∞}^{∞} so that there is a subsequence n_{j} with

a_{n_{j}+ℓ} → a_{ℓ}^{(r)} b_{n_{j}+ℓ} → b_{ℓ}^{(r)}  (1.7)

as j → ∞ for each fixed ℓ = 0, ±1, ±2, . . .. In [48], we noted that

Proposition 1.6. For each right limit point, σ(J^{(r)}) ⊂ σ_{ess}(J).

This is a basic result that many, including us, regard as immediate. For if λ ∈ σ(J^{(r)}) and φ^{(m)} is a sequence of unit trial functions with ∥(J^{(r)} − λ)φ^{(m)}∥ → 0, then for any j(m) → ∞, ∥(J − λ)φ^{(m)}(· + n_{j(m)})∥ → 0, and if j(m) is chosen going to infinity fast enough, then φ^{(m)}(· − n_{j(m)}) → 0 weakly, so λ ∈ σ_{ess}(J).

Let R be the set of right limit points. Clearly, Proposition 1.6 says that

\overline{⋃_{r∈R} σ(J^{(r)})} ⊂ σ_{ess}(J)  (1.8)

Our new realization here for this example is that

Theorem 1.7. If (1.6) holds, then

\overline{⋃_{r∈R} σ(J^{(r)})} = σ_{ess}(J)  (1.9)

Remark. It is an interesting question whether anything is gained in (1.9) by taking the closure—that is, whether the union is already closed. In every example we can analyze the union is closed. V. Georgescu has informed us that the methods of [26] imply that the union is always closed and that the details of the proof of this fact are the object of a paper in preparation

Surprisingly, the proof will be a rather simple trial function argument. The difficulty with such an argument tried naively is the following: To say J^{(r)} is a right limit point means that there are L_{m} → ∞ so that J ↾ [n_{j(m)} − L_{m}, n_{j(m)} + L_{m}] shifted to [−L_{m}, L_{m}] converges uniformly to J^{(r)} ↾ [−L_{m}, L_{m}]. But L_{m} might grow very slowly with m. Weyl’s criterion says that if λ ∈ σ_{ess}(J), there are trial functions, φ_{k}, supported on [n_{k} − L̃_{k}, n_{k} + L̃_{k}] so ∥(J − λ)φ_{k}∥ → 0. By a compactness argument, one can suppose the n_{k} are actually n_{j(m)}’s for some right limit. The difficulty is that L̃_{m} might grow much faster than L_{m}, so translated φ_{k}’s are not good trial functions for J^{(r)}.

The key to overcoming this difficulty is to prove that one can localize trial functions in some interval of fixed size L, making a localization error of O(L^{−1}). This is what we will do in Section 2. In this idea, we were motivated by arguments in Avron et al. [6], although to handle the continuum case, we will need to work harder.

The use of localization ideas to understand essential spectrum, an implementation using double commutators, is not new—it goes back to Enss [22] and was raised to high art by Sigal [73]. Enss and Sigal, and also Agmon [1] and Garding [24], later used these ideas and positivity inequalities to locate inf σ_{ess}(H), which suffices for the HVZ theorem but not for some of our applications.

What distinguishes our approach and allows stronger results is that, first, we use trial functions exclusively and, second, as noted above, we study all of σ_{ess} rather than only its infimum. Third, and most significantly, we do not limit ourselves to sets that are cones near infinity and instead take balls. This gives us small operator errors rather than compact operator errors (although one can modify our arguments and take ball sizes that go to infinity slowly, and so get a compact localization error). It makes the method much more flexible.

While this paper is lengthy because of many different applications, the underlying idea is captured by the mantra “localization plus compactness.” Here compactness means that resolvents restricted to balls of fixed size translated to zero lie in compact sets. We have in mind the topology of norm convergence once resolvents are multiplied by the characteristic functions of arbitrary fixed balls.

Because we need to control ∥(A−λ)φ∥^{2} and not just ⟨φ, (A−λ)φ⟩, if we used double commutators, we would need to control [j, [j, (A−λ)^{2}]], so in the continuum case we get unbounded operators and the double commutator is complicated. For this reason, following [6] and [35], we use single commutators and settle for an inequality rather than the equality one gets from double commutators.

After we completed this paper and released a preprint, we learned of some related work using C^{∗}-algebra techniques to compute σ_{ess}(H) as the closure of a union of spectra of asymptotic Hamiltonians at infinity; see Georgescu-Iftimovici [26] and Mantoiu [55]. Further work is in [4, 25, 27, 28, 56, 70].

We also learned of very recent work of Rabinovich [66], based in part on [47, 61, 64, 65, 67, 68], using the theory of Fredholm operators to obtain results on essential spectrum as a union of spectra of suitable limits at infinity.

Thus, the notion that in great generality the essential spectra is a union of spectra of limits at infinity is not new. Our contributions are twofold: First, some may find our direct proof via trial functions more palatable than arguments relying on considerable machinery. Second, our examples of Section 4, Section 5, and Section 7(b),(c) are, so far as we know, new, although it is certainly true that the methods of [26, 55, 66], can analyze some or even all these examples. In particular, we settle Conjecture 12.2.3 of [77].

There is obviously considerable overlap in philosophies (which, after all, both extend the ideas of the HVZ theorem) and results. The techniques seem to be rather different, although we suspect a translation of the C^{∗}-algebra machinery to more prosaic terms will show similarities that are, for now, not clear to us.

The paper [27] has results stated without reference to C^{∗}-algebras (although the proofs use them) and, in particular, our Theorems 3.7 and 3.12 are special cases of Theorem 1.1 of [27].

We present the localization lemmas in Section 2 and prove our main results in Section 3. Section 4 discusses an interesting phenomena involving Schrödinger operators with severe oscillations at infinity. Section 5 has the applications to potentials asymptotic to isospectral tori and includes results stronger than Theorems 1.2, 1.3, and 1.5. In particular, we settle positively Conjecture 12.2.3 of [77]. Section 6 discusses the HVZ theorem, and Section 7 other applications. Section 8 discusses magnetic fields.

We can handle the common Schrödinger operators associated to quantum theory with or without magnetic fields as well as orthogonal polynomials on the real line (OPRL) and unit circle (OPUC).

It is a pleasure to thank D. Damanik and R. Killip for useful discussions, and V. Georgescu, M. Mantoiu, V. Rabinovich, A. Sobolev and B. Thaller for useful correspondence. This research was completed during B. S.’s stay as a Lady Davis Visiting Professor at The Hebrew University of Jerusalem. He would like to thank H. Farkas for the hospitality of the Einstein Institute of Mathematics at The Hebrew University.

2. Localization Estimates

Here we will use localization formulae but with partitions of unity that are concentrated on balls of fixed size in place of the previous applications that typically take j’s that are homogeneous of degree zero near infinity. Also, we use single commutators.

Let H be a separable Hilbert space and A a selfadjoint operator on H. Let {j_{α}} be a set of bounded selfadjoint operators indexed by either a discrete set, S, like ℤ^{ν} or by α ∈ ℝ^{ν}. In the latter case, we suppose j_{α} is measurable and uniformly bounded in α. We assume that {j_{α}} is a partition of unity, namely,

∑_{α∈S} j_{α}^{2} = 1 or  ∫_{α∈ℝ^{ν}} j_{α}^{2} d^{ν}α = 1  (2.1)

where the convergence of the sum or the meaning of the integral is in the weak operator topology sense. Two examples that will often arise are where H = ℓ^{2}(ℤ^{ν}), ψ ∈ ℓ^{2}(ℤ^{ν}) is real-valued with ∑_{n} ψ(n)^{2} = 1, and {j_{m}}_{m∈ℤ^{ν}} is multiplication by ψ(·−m), or where H = L^{2}(ℝ^{ν}, d^{ν}x), ψ ∈ L^{2}(ℝ^{ν}, d^{ν}x)∩L^{∞}(ℝ^{ν}, d^{ν}x) is real-valued with ∫ ψ(x)^{2} d^{ν}x = 1, and {j_{y}}_{y∈ℝ^{ν}} is multiplication by ψ(· − y).

Assume that for each α, j_{α} maps the domain of A to itself and let φ be a vector in the domain of A. Notice that

∥Aj_{α}φ∥^{2} = ∥(j_{α}A + [A, j_{α}])φ∥^{2}
≤ 2∥j_{α}Aφ∥^{2} + 2∥[A, j_{α}]φ∥^{2}  (2.2)

Thus

Proposition 2.1.

∑_{α} ∥Aj_{α}φ∥^{2} ≤ 2∥Aφ∥^{2} + ⟨φ, Cφ⟩  (2.3)

where

C = 2 ∑_{α} −[A, j_{α}]^{2}  (2.4)

Remark. Since [A, j_{α}] is skew-adjoint, −[A, j_{α}]^{2} = [j_{α}, A]^{∗}[j_{α}, A] ≥ 0.

Proof. (2.3) is immediate from (2.2) since

∑_{α} ∥j_{α}Aφ∥^{2} = ∑_{α}⟨Aφ, j_{α}^{2}Aφ⟩ = ∥Aφ∥^{2}  (2.5)

and

∥[A, j_{α}]φ∥^{2} = −⟨φ, [A, j_{α}]^{2}φ⟩  (2.6)
□

Theorem 2.2. There exists an α so that j_{α}φ ≠ 0 and

∥Aj_{α}φ∥^{2} ≤ \{2(\frac{∥Aφ∥}{∥φ∥})^{2} + ∥C∥\}∥j_{α}φ∥^{2}  (2.7)

Proof. Call the quantity in { } in (2.7) d. Then, since ∥φ∥^{2} = ∑_{α} ∥j_{α}φ∥^{2}, (2.3) implies

∑_{α} [∥Aj_{α}φ∥^{2} − d∥j_{α}φ∥^{2}] ≤ 0

so at least one term with ∥j_{α}φ∥ ≠ 0 is nonpositive. □

To deal with unbounded A’s, we will want to suppose that \sqrt{C} is A-bounded:

Theorem 2.3. Suppose A is unbounded and

⟨φ, Cφ⟩ ≤ δ(∥Aφ∥^{2} + ∥φ∥^{2})  (2.8)

Then there is an α with j_{α}φ ≠ 0 so that

∥Aj_{α}φ∥^{2} ≤ \{(2 + δ) \frac{∥Aφ∥^{2}}{∥φ∥^{2}} + δ\}∥j_{α}φ∥^{2}  (2.9)

Proof. By (2.3) and (2.8), we have

∑_{α} ∥Aj_{α}φ∥^{2} ≤ (2 + δ)∥Aφ∥^{2} + δ∥φ∥^{2}

so, as before, (2.9) follows. □

3. The Essential Spectrum

This is the central part of this paper. We begin with Theorem 1.7, the simplest of the results:

Proof of Theorem 1.7. We already proved (1.8) in the remarks after Proposition 1.6, so suppose λ ∈ σ_{ess}(J). Recall Weyl’s criterion, λ ∈ σ_{ess}(J) ⇔ there exist unit vectors φ_{m} ⟶^{w} 0 with ∥(J − λ)φ_{m}∥ → 0.

Given ε, pick a trial sequence {φ_{m}}, such that each φ_{m} is supported in {n | n > m}, so that

∥(J − λ)φ_{m}∥^{2} ≤ \frac{1}{3} ε^{2}∥φ_{m}∥^{2}  (3.1)

which we can do, by Weyl’s criterion, since f_{j} ⟶^{w} 0 implies ∑_{n<m}|f_{j}(n)|^{2} → 0 for each m.

For L = 1, 2, 3, …, let

ψ_{L}(n) = \begin{cases}\frac{n−1}{L} & n = 1, 2, …, L \\ \frac{2L−1−n}{L} & n = L, L + 1, …, 2L − 1 \\ 0 & n ≥ 2L − 1 \end{cases}  (3.2)

and let

c_{L}^{2} = ∑_{n}|ψ_{L}(n)|^{2}  (3.3)

so that c_{L} ∼ L^{1/2} in the sense that for some 0 < a ≤ b < ∞,

aL^{1/2} ≤ c_{L} ≤ bL^{1/2}  (3.4)

For α = 1, 2, …, let

j_{α,L}(n) = c_{L}^{−1}ψ_{L}(n + α)  (3.5)

so, by (3.3),

∑_{α} j_{α,L}^{2} ≡ 1  (3.6)

Since |ψ_{L}(n + 1) − ψ_{L}(n)| ≤ L^{−1}, we see that

|⟨δ_{n}, [j_{α,L}, J]δ_{m}⟩| = \begin{cases}sup_{n}|a_{n}| c_{L}^{−1}L^{−1} & if |n − m| = 1 and |n − α − L| ≤ L \\ 0 & otherwise \end{cases}

(3.7)

Therefore, C ≡ ∑_{α} 2[j_{α,L}, J]^{2} is a 5-diagonal matrix with matrix elements bounded by

2 · 2(2L)c_{L}^{−2}L^{−2}( sup_{n} |a_{n}|)^{2}  (3.8)

where the second two comes from the number of k’s that make a nonzero contribution to ⟨δ_{n}, [j_{α,L}, J]δ_{k}⟩⟨δ_{k}, [j_{α,L}, J]δ_{m}⟩. By (3.4), there is a constant K depending on sup_{n}|a_{n}| so that

∥C∥ ≤ KL^{−2}  (3.9)

Picking L so KL^{−2} < ε^{2}/3, we see, by Theorem 2.2, there is a j_{α_{m}} so ∥j_{α_{m}}φ_{m}∥ ≠ 0 and

∥(J − λ)j_{α_{m}}φ_{m}∥ ≤ ε∥j_{α_{m}}φ_{m}∥  (3.10)

The intervals

I_{m} = [α_{m} + 1, α_{m} + 2L − 1]

which support j_{α_{m}}φ_{m}, have fixed size, and move out to infinity since I_{m} ⊂ {n | n ≥ m − L}. Since the set of real numbers with |b| + |a| + |a|^{−1} ≤ K is compact and L is finite, we can find a right limit point J^{(r)} so that a subsequence of J ↾ I_{m} translated by α_{m} + L converges to J^{(r)} ↾ [1 − L, L − 1]. Using translations of the trial functions j_{α_{m}}φ_{m}, we find ψ_{m} so

lim_{m→∞} \frac{∥(J^{(r)} − λ)ψ_{m}∥}{∥ψ_{m}∥} ≤ ε  (3.11)

which means

dist(λ, σ(J^{(r)})) ≤ ε  (3.12)

Since ε is arbitrary, we have λ ∈ \overline{∪σ(J^{(r)})}. □

We have been pedantically careful about the above proof so that below we can be much briefer and just relate to this idea as “localization plus compactness” and not provide details.

We turn next to the CMV matrices defined by a sequence of Verblunsky coefficients {α_{j}}_{j=0}^{∞} with α_{j} ∈ 𝔻. We define the unitary 2×2 matrix

Θ(α) = ( \begin{matrix}\overline{α} & ρ \\ ρ & −α \end{matrix} ) where ρ = (1 − |α|^{2})^{1/2} and L = Θ_{0} ⊕ Θ_{2} ⊕ Θ_{4} ⊕ ⋯,

M = 1 ⊕ Θ_{1} ⊕ Θ_{3} + ⋯, where 1 is a 1 × 1 matrix and Θ_{j} = Θ(α_{j}). Then the CMV matrix is the unitary matrix C = LM. Given a two-sided sequence {α_{j}}_{j=−∞}^{∞}, we define L̃ = ⋯ Θ_{−2} ⊕ Θ_{0} ⊕ Θ_{2} and M̃ = Θ_{−1} ⊕Θ_{1} ⊕Θ_{3} ⊕⋯ on ℓ^{2}(ℤ) where Θ_{j} acts on the span of δ_{j} and δ_{j+1}. We set C̃ = L̃M̃. See [76, 77] for a discussion of the connection of CMV and extended CMV matrices to OPUC.

In [76, 77], C̃ is used for the transpose of C (alternate CMV matrix). Its use here is very different!

If {α_{j}}_{j=0}^{∞} is a set of Verblunsky coefficients with

sup_{j} |α_{j}| < 1  (3.13)

we call {β_{j}}_{j=−∞}^{∞} a right limit point if there is a sequence m_{j} so that for ℓ = 0, ±1, …,

lim_{j→∞} α_{m_{j}+ℓ} = β_{ℓ}  (3.14)

and we call C̃(β) a right limit of C(α). We have

Theorem 3.1. Let C(α) be the CMV matrix of a sequence obeying (3.13). Let R be the set of right limit extended CMV matrices. Then

σ_{ess}(C(α)) = \overline{⋃_{R} C̃(β)}  (3.15)

Proof. The arguments of Section 2 extend to unitary A if −[j_{α}, A]^{2} is replaced by [j_{α}, A]^{∗}[j_{α}, A]. Matrix elements of [j_{α}, C] are bounded by sup_{n,|k|≤2}|j_{α}(n + k) − j_{α}(n)| since C has matrix elements bounded by 1 and is 5-diagonal. Thus, C is 9-diagonal, but otherwise the argument extends with no change since {α | |α| ≤ sup_{j}|α_{j}|} is a compact subset of 𝔻. □

Next, we want to remove the condition that sup|α_{j}| < 1 in the OPUC case and the conditions sup|b_{j}| < ∞ and inf|a_{j}| > 0 in the OPRL case. The key, of course, is to preserve compactness, that is, existence of limit points, and to do that, we need only extend the notion of right limit.

If \{α_{j}\}_{j=−∞}^{∞} is a two-sided sequence in \overline{𝔻}, one can still define C̃(α_{j}) since Θ(α_{j}) makes sense. If |α_{j}| = 1, then ρ_{j} = 0 and Θ(α_{j}) = ( \begin{matrix}\overline{α}_{j} & 0 \\ 0 & −α_{j} \end{matrix} ) is a direct sum in such a way L and M both decouple into direct sums on ℓ^{2}(−∞, j] ⊕ ℓ^{2}[j + 1, ∞) so C decouples. If a single α_{j} has |α_{j}| = 1, we decouple into two semi-infinite matrices (both related by unitary transforms to ordinary CMV matrices), but if more than one α_{j} has |α_{j}| = 1, there are finite direct summands.

In any event, we can define C̃(α_{j}) for \{α_{j}\} ∈ ×_{j=−∞}^{∞} \overline{𝔻} and define right limit points of C(α_{j}) even if sup|α_{j}| = 1. Since matrix elements of C are still bounded by 1, C is still 5-diagonal and ×_{j=−∞}^{∞} \overline{𝔻} is compact, we immediately have

Theorem 3.2. With the extended notion of C̃, Theorem 3.1 holds even if (3.13) fails.

For bounded Jacobi matrices, we still want sup(|a_{n}| + |b_{n}|) < ∞, but we do not need inf|a_{n}| > 0. Again, the key is to allow two-sided Jacobi matrices, J_{r}, with some a_{n} = 0, in which case J_{r} decouples on ℓ^{2}(−∞, n] ⊕ ℓ^{2}[n + 1, ∞). If a single a_{n} = 0, there are two semi-infinite matrices. If more than one a_{n} = 0, there are finite Jacobi summands. Again, with no change in proof except for the change in the meaning of right limits to allow some a_{n}^{(r)} = 0, we have

Theorem 3.3. Theorem 1.7 remains true if (1.6) is replaced by

sup_{n} (|a_{n}| + |b_{n}|) < ∞  (3.16)

so long as J^{(r)} are allowed with some a_{n}^{(r)} = 0.

In Section 7, we will use Theorems 3.2 and 3.3 to complement the analysis of Krein (which appeared in Akhiezer-Krein [3]) for bounded Jacobi matrices with finite essential spectrum, and of Golinskii [29] for OPUC with finite derived sets.

Our commutator argument requires that |a_{n}| is bounded, but one can also handle lim sup|b_{n}| = ∞. It is useful to define:

Definition. Let A be a possibly unbounded selfadjoint operator. We say that +∞ lies in σ_{ess}(A) if σ(A) is not bounded above, and −∞ lies in σ_{ess}(A) if σ(A) is not bounded below.

We now allow two-sided Jacobi matrices, J̃, with b_{n} = +∞ and/or b_{n} = −∞ (and also a_{n} = 0). If |b_{n}| = ∞, we decouple into ℓ^{2}(−∞, n − 1]⊕ℓ^{2}[n+1, ∞) and place b_{n} in “σ_{ess}(J̃).” With this extended definition, we still have compactness, that is, for any intervals in ℤ_{+}, I_{1}, I_{2}, … of fixed finite size, ℓ, with ℓ^{−1} ∑_{j∈I_{n}} j → ∞, there is a subsequence converging to a set of Jacobi parameters with possibly b_{n} = +∞ or b_{n} = −∞. We therefore have Theorem 3.4. Theorem 1.7 remains true if (1.6) is replaced by

sup_{n} |a_{n}| < ∞  (3.17)

so long as J^{(r)} are allowed to have some a_{n}^{(r)} = 0 and/or some b_{n}^{(r)} = ±∞.

Remarks. 1. This includes the conventions on when ±∞ lies in σ_{ess}(J). To prove this requires a simple separate argument. Namely, ⟨δ_{n}, Jδ_{n}⟩ = b_{n}, so b_{n} ∈ numerical range of J = convex hull of σ(J). Thus, if b_{n_{j}} → ±∞, then ±∞ ∈ σ(J).

2. If sup_{n}|a_{n}| = ∞, σ_{ess} can be very subtle; see [42, 43].

Next, we turn to Jacobi matrices on ℤ^{ν} (including ν = 1), that is, J acts on ℓ^{2}(ℤ^{ν}) via

(Ju)(n) = ∑_{|m−n|=1} a_{(n,m)}u(m) + ∑_{n} b_{n}u(n)  (3.18)

where the b_{n}’s are indexed by n ∈ ℤ^{ν} and the a_{(n,m)}’s by bonds {m, n} (unordered pairs) with |m − n| = 1. For simplicity of exposition, we suppose

sup_{|m−n|=1} (|a_{(n,m)}| + |a_{(n,m)}|^{−1}) + sup_{n} |b_{n}| < ∞  (3.19)

although we can, as above, also handle some limits with a_{(n,m)} = 0 or some |b_{n}| = ∞. With no change, one can also control finite-range off-diagonal terms, and with some effort on controlling [j_{α}, J], it should be possible to control infinite-range off-diagonal terms with sufficiently rapid off-diagonal decay.

Let us call J̃ a limit point of J at infinity if and only if there are points n_{j} ∈ ℤ^{ν} with n_{j} → ∞ so that for every finite k, ℓ,

b_{n_{j}+ℓ} → b̃_{ℓ} a_{(n_{j}+ℓ,n_{j}+k)} → ã_{(k,ℓ)}  (3.20)

Let L denote the set of limits J̃. Then

Theorem 3.5. Let J be a Jacobi matrix of the form (3.18) on ℓ^{2}(ℤ^{ν}). Suppose (3.19) holds. Then

σ_{ess}(J) = \overline{⋃_{J̃∈L} σ(J̃)}  (3.21)

Proof. We can define partitions of unity j_{α,L} indexed by α ∈ ℤ^{ν} with j_{α}(n) ≠ 0 only if |n − α| ≤ L and with − ∑_{α}[j_{α}, J]^{2} bounded by O(L^{−2}). With this, the proof is the same as in the one-dimensional case. □

It is often comforting to only consider limit points in a single direction. Because the sphere is compact, this is easy.

Definition. Let e ∈ S^{ν−1}, the unit sphere in ℝ^{ν}. We say J̃ is a limit point in direction e if the n_{j} in (3.20) obey n_{j}/|n_{j}| → e. We let L_{e} denote the limit points in direction e.

Suppose J̃ is a limit point for J with sequence n_{j}. Since S^{ν−1} is compact, we can find a subsequence n_{j(k)} so n_{j(k)}/|n_{j(k)}| → e_{0} for some e_{0}. The subsequence also converges to J̃ so J̃ is a limit point for direction e_{0}. Thus,

Theorem 3.6. Let J be a Jacobi matrix of the form (3.18) on ℓ^{2}(ℝ^{ν}). Suppose (3.19) holds. Then

σ_{ess}(J) = \overline{⋃_{e∈S^{ν−1}} ⋃_{J̃∈L_{e}} σ(J̃)}  (3.22)

For example, if ν = 1, we can consider left and right limit points. Finally, we turn to Schrödinger operators. Here we need some kind of

compactness condition of the −Δ + V that prevents V from oscillating wildly at infinity (but see the next section). We begin with a warmup case that will be the core of our general case:

Theorem 3.7. Let V be a uniformly continuous, bounded function on ℝ^{ν}. For each e ∈ S^{ν}, call W a limit of V in direction e if and only if there exists x_{j} ∈ ℝ^{ν} with |x_{j}| → ∞ and x_{j}/|x_{j}| → e so that V (x_{j} + y) → W (y). Then, with L_{e} the limits in direction e,

σ_{ess}(−Δ + V) = \overline{⋃_{e} ⋃_{W ∈L_{e}} σ(−Δ + W)}  (3.23)

Remarks. 1. While we have not stated it explicitly, there is a result for half-line operators.

2. Uniform continuity means ∀ε, ∃δ, so |x−y| < δ ⇒ |V (x)−V (y)| < ε. It is not hard to see this is equivalent to {V (· + y)}_{y∈ℤ^{ν}} being equicontinuous.

3. This result appears in a more abstract formulation in Georgescu-Iftimovici [27].

Proof. As noted, uniform continuity implies uniform equicontinuity so, by the Arzela-Ascoli theorem (see [69]), given any sequence of balls {x | |x − y_{j}| ≤ L}, there is an e and a W in L_{e} so V (· + y_{j}) → W (·) uniformly on {x | |x| ≤ L}. This is the compactness needed for our argument.

To handle localization, pick any nonnegative rotation invariant C^{∞} function ψ supported on {x | |x| ≤ 1} with ∫ ψ(x)^{2} d^{ν}x = 1. Define j_{x,L} as the operator of multiplication by the function

j_{x,L}(y) = L^{−ν/2}ψ(L^{−1}(y − x))

and note that

∫ j_{x,L}^{2} d^{ν}x = 1

With A = (−Δ + V − λ) and C = 2 ∫ −[A, j_{x,L}]^{2} d^{ν}x, we have (2.8) with δ = O(L^{−2}), since C = L^{−2}(c_{1}Δ + c_{2}) for constants c_{1} and c_{2} (for C is translation and rotation invariant and scale covariant).

(3.23) follows in the usual way. □

Our final result in this section concerns Schrödinger operators with potentially singular V ’s. As in the last case, we will suppose regularity at infinity. In the next section, we will show how to deal with irregular oscillations near infinity. Recall the Kato class and norm [2, 19] is defined by

Definition. V : ℝ^{ν} → ℝ is said to live in the Kato class, K_{ν}, if and only if

lim_{α↓0} [ sup_{x} ∫_{|x−y|≤α}|x − y|^{2−ν}|V (y)| d^{ν}y] = 0  (3.24)

(If ν = 1, 2, the definition is different. If ν = 2, |x − y|^{2−ν} is replaced by log[|x − y|^{−1}], and if ν = 1, we require sup_{x} ∫_{|x−y|≤1}|V (y)| dy < ∞.) The K_{ν} norm is defined by

∥V ∥_{K_{ν}} = sup_{x} ∫_{|x−y|≤1}|x − y|^{2−ν}|V (y)| d^{ν}y  (3.25)

We introduce here

Definition. V : ℝ^{ν} → ℝ is called uniformly Kato if and only if V ∈ K_{ν} and

lim_{y↓0} ∥V − V (· − y)∥_{K_{ν}} = 0  (3.26)

Example 3.8. Let

V (x) = sin(x_{1}^{2})  (3.27)

Then V ∈ K_{ν}, but for (x_{0})_{1} large and y = (π/2(x_{0})_{1}, y_{2}, …), [(x_{0} + y)_{1}]^{2} = (x_{0})_{1}^{2}+π+O(1/(x_{0})_{1}), so for x near x_{0}, V (x)−V (x−y) ∼ 2V (x), and because of the |V (·)| in (3.25), we do not have (3.26). We discuss this further in the next section.

Example 3.9. We say p is canonical for ℝ^{μ} if p = μ/2 where μ ≥ 3, p > 2 if μ = 2, and p = 1 if μ = 1. If

sup_{x} ∫_{|x−y|≤1}|V (y)|^{p} d^{μ}y < ∞  (3.28)

then V ∈ K_{μ} (see [19]). Moreover, if

lim_{|x|→∞} ∫_{|x−y|≤1}|V (y)|^{p} d^{μ}y = 0  (3.29)

it is easy to see that (3.26) holds because V is small at infinity, and (3.26) holds for L^{p} norm if V ∈ L^{p}.

Example 3.10. If π: ℝ^{ν} → ℝ^{μ} is a linear map onto ℝ^{μ} and W ∈ K_{μ}, then V (x) = W (πx) is in K_{ν} and the K_{ν} norm of V is bounded by a π-dependent constant times the K_{μ} norm of W. If W obeys (3.26), so does V.

We will combine Examples 3.9 and 3.10 in our study of the HVZ theorem.

Proposition 3.11. Let V be a uniformly Kato potential on ℝ^{ν} and let H_{x} = −Δ + (· − x). Then for any sequence x_{k} → ∞, there is a subsequence x_{k(m)} and a selfadjoint operator H_{∞} so that for z ∈ ℂ\[a, ∞) for some a ∈ ℝ, we have

∥[(H_{x_{k(m)}} − z)^{−1} − (H_{∞} − z)^{−1}]χ_{S}∥ → 0  (3.30)

for χ_{S}, the characteristic function of an arbitrary bounded set.

Remark. Formally, H_{∞} is a Schrödinger operator of the form H_{0} + V_{∞}, but V_{∞}, as constructed, is only in the completion of K_{ν}, and that is known to include some distributions (see [30, 54]).

Proof. It is known that if W ∈ K_{ν}, then W is −Δ form bounded with relative bound zero with bounds depending only on K_{ν} norms (see [19]). Thus, since all V_{x}’s have the same K_{ν} norm, we can find a so H_{x} ≥ a for all x. It also means that for each z ∈ ℂ\[a, ∞), we can bound ∥|W |^{1/2}(H_{x} − z)^{−1}Δ^{1/2}∥ by c∥W ∥_{K_{ν}} with c only z-dependent and ∥V ∥_{K_{ν}}-dependent.

Let φ be a C^{∞} function of compact support and note (constants are z- or ∥V ∥_{K_{ν}}-dependent)

∥|W |^{1/2}[(H − z)^{−1}, φ]∥ ≤ ∥W^{1/2}(H − z)^{−1}[Δ, φ](H − z)^{−1}∥
≤ c∥W^{1/2}(H − z)^{−1}Δ^{1/2}∥ ∥∇φ∥

This in turn implies that if S_{1} is a ball of radius r fixed about x_{0} and S_{2} a ball of radius R > r, then

∥W^{1/2}(1 − χ_{S_{2}})(H − z)^{−1}χ_{S_{1}}∥ → 0

as R → ∞. So if ∥(W_{n} − W)χ_{S}∥_{K_{ν}} → 0 for all balls, and sup_{n} ∥W_{n}∥_{K_{ν}} < ∞, then

∥((−Δ + W_{n} − z)^{−1} − (−Δ + W − z)^{−1})χ_{S}∥ → 0

for all S.

In this way, we see that if V is uniformly Kato and V_{x_{n}} → V_{∞} in K_{ν} uniformly on all balls, then

∥[(H_{x_{n}} − z)^{−1} − (H_{∞} − z)]^{−1}χ_{S}∥ → 0  (3.31)

The condition of V being uniformly Kato means convolutions of V with a C^{∞} approximate identity converge to V in K_{ν} norm. Call the approximations V^{(m)}. Each is C^{∞} with bounded derivatives and so, by the equicontinuity argument in Theorem 3.7, we can find x_{j_{m}(n)} and V_{∞}^{(m)} so

∥[(−Δ + V_{x_{jm(n)}}^{(m)} − z)^{−1} − (−Δ + V_{∞}^{(m)} − z)^{−1}]χ_{S}∥ → 0

Since V_{x}^{(m)} → V_{x} uniformly in x, a standard ε/3 argument (see [69]) shows that one can find x_{j(m)} so ∥[(H_{x_{m}} − z)^{−1} − (H_{x_{m′}} − z)^{−1}]χ_{S}∥ is small for each S as m, m′ → ∞. In this way, we obtain the necessary limit operator. □

Given V uniformly Kato, the limits constructed by Proposition 3.11 where x_{n}/|x_{n}| → e are called limits of H in direction e. Again, the next result appears in a more abstract setting in Georgescu-Iftimovici [27].

Theorem 3.12. Let V be uniformly Kato. Let L_{e} denote the limits of H in direction e. Then,

σ_{ess}(H) = \overline{⋃_{e} ⋃_{H_{∞}∈L_{e}} σ(H_{∞})}  (3.32)

The papers that use C^{∗}-algebras [26, 55] study h(p) + V in place of −Δ + V. These papers only required that h(p) → ∞ as p → ∞. It seems likely that for many such h’s, our methods will work. h(p) → ∞ as p → ∞ is critical in our approach to assure that if φ_{n} → 0 weakly and ∥(H − E)φ_{n}∥ → 0 then χ_{{x||x|<R}}φ_{n} → 0.

It is likely that one can develop a theory for h(p) + V (x) without supposing h(p) → ∞ or even f(p, x), but one would need to consider limits at infinity in phase space, not just on configuration space.

Proof. We pick a so H_{x} ≥ a for all x. Pick z ∈ (−∞, a) and let Ã_{x} = (H_{x} − z)^{−1}. As above, ∥[Ã_{x}, j_{α}]∥ ≤ c∥∇j_{α}∥ for any j_{α} in C_{0}^{∞}. For λ ∈ σ_{ess}(H), let A = (H_{x} − z)^{−1} − (λ − z)^{−1}. Theorem 2.2 provides the necessary localization estimate. Proposition 3.11 provides the necessary compactness. (3.32) is then proven in the same way as earlier theorems. □

4. Schrödinger Operators With Severe Oscillations at Infinity

This section is an aside to note that the lack of uniformity at infinity that can occur if V is merely K_{ν} is irrelevant to essential spectrum. We begin with Example 3.8, the canonical example of severe oscillations at infinity:

Proposition 4.1. Let

W (x) = sin(x^{2})  (4.1)

on (0, ∞) and let H_{0} = −\frac{d^{2}}{dx^{2}} with u(0) = 0 boundary conditions. Then (1) W (H_{0} + 1)^{−1} is not compact.

(2) (H_{0} + 1)^{−1/2}W (H_{0} − 1)^{−1/2} is compact.

Remarks. 1. Our proof of (1) shows that W f(H_{0}) is noncompact for any continuous f ≢ 0 on (0, ∞).

2. Consideration of W = ∇ · Q⃗ potentials goes back to the 1970’s; (see [8, 11, 12, 16, 17, 20, 39, 40, 57, 71, 72, 78]).

Proof. (1) Let φ be a nonzero C_{0}^{∞}(0, ∞) function in L^{2} and let

ψ_{n}(x) = [(H_{0} + 1)φ](x − n)  (4.2)

Then

∥W (H_{0} + 1)^{−1}ψ_{n}∥^{2} = ∫ W (x)^{2}φ(x − n)^{2} dx
= \frac{1}{2} ∫ φ(x)^{2} dx − \frac{1}{2} ∫ cos(2x^{2})φ(x − n)^{2} dx
→ \frac{1}{2} ∫ φ(x)^{2} dx ≠ 0  (4.3)

by an integration by parts. Since ψ_{n} ⟶^{w} 0, this shows W (H_{0} − 1)^{−1} is not compact.

(2) Since \frac{d}{dx}[−\frac{1}{2x} cos(x^{2})] = sin(x^{2}) + O(x^{−2}), we see Q(x) = lim_{y→∞} − ∫_{x}^{y} W (z) dz exists and obeys

|Q(x)| ≤ c(x + 1)^{−1}  (4.4)

Thus W = [\frac{d}{dx}, Q], so

(H_{0} + 1)^{−1/2}W (H_{0} + 1)^{−1/2} = ((H_{0} + 1)^{−1/2} \frac{d}{dx})(Q(H_{0} + 1)^{−1/2}) + cc

Since (H_{0} + 1)^{−1/2} \frac{d}{dx} is bounded and Q(H_{0} − 1)^{−1/2} is compact (by (4.4)), (H_{0} + 1)^{−1/2}W (H_{0} + 1)^{−1/2} is compact. □

Thus, oscillations at infinity are irrelevant for essential spectrum! While the slick argument above somewhat obscures the underlying physics, the reason such oscillations do not matter has to do with the fact that σ_{ess}(H) involves fixed energy, and oscillations only matter at high energy. Our proof below will implement this strategy more directly.

We begin by noting that the proof of Proposition 3.11 implies the following:

Theorem 4.2. Suppose V_{n} is a sequence of multiplicative operators so that

(i) For any ε > 0, there is C_{ε} so that

⟨φ, |V_{n}|φ⟩ ≤ ε∥∇φ∥^{2} + C_{ε}∥φ∥^{2}  (4.5)

for any n and all φ ∈ Q(−Δ). (ii) For any ball S about zero,

∥(−Δ + 1)^{−1/2}χ_{S}(V_{n} − V_{m})(−Δ + 1)^{−1/2}∥ → 0  (4.6)

as n, m → ∞. Then for any ball and z ∈ ℂ\[a, ∞),

∥[(−Δ + V_{n} − z)^{−1} − (−Δ + V_{m} − z)^{−1}]χ_{S}∥ → 0  (4.7)

Moreover, if (4.6) holds as n → ∞ with V_{m} replaced by some V_{∞}, then

lim_{n→∞} ∥[(−Δ + V_{n} − z)^{−1} − (−Δ + V_{∞} − z)^{−1}]χ_{S}∥ = 0  (4.8)

As an immediate corollary, we obtain

Theorem 4.3. Let V ∈ K_{ν} obey

lim_{R→∞} sup_{|x|≥R} ∫_{|x−y|≤1}|x − y|^{−(ν−2)}|V (y)| d^{ν}y = 0  (4.9)

Then

σ_{ess}(−Δ + V) = [0, ∞)  (4.10)

Remark. If (4.9) holds, we say that V is K_{ν} small at infinity.

Proof. By Theorem 4.2, if x_{n} → ∞,

∥[(−Δ + V (· − x_{n}) − z)^{−1} − (−Δ − z)^{−1}]χ_{S}∥ → 0  (4.11)

so, in a sense, −Δ is the unique limit point at infinity. The standard localization argument proves (4.10). □

Here is the key to studying general V ∈ K_{ν} with no uniformity at infinity:

Proposition 4.4. Let V_{n} be a sequence of functions supported in a fixed ball {x | |x| ≤ R}. Suppose

lim_{α↓0} sup_{n,x} ∫_{|x−y|≤α}|x − y|^{−(ν−2)}|V_{n}(y)| d^{ν}y = 0  (4.12)

Then there is a subsequence V_{n(j)} so

lim_{j,k→∞} ∥(−Δ + 1)^{−1/2}(V_{n(j)} − V_{n(k)})(−Δ + 1)^{−1/2}∥ = 0  (4.13)

Proof. Given K, let P_{K} be the projection in momentum space onto |p| ≤ K and Q_{K} = 1 − P_{K}. (4.12) implies that for any ε > 0,

⟨φ, |V_{n}|φ⟩ ≤ ε∥∇φ∥^{2} + C_{ε}∥φ∥^{2}  (4.14)

for a fixed C_{ε} and all n. This implies that

∥|V_{n}|^{1/2}(−Δ + 1)^{−1/2}Q_{K}∥^{2} ≤ ε + C_{ε}(K^{2} + 1)^{−1/2}  (4.15)

so

lim_{K→∞} sup_{n} ∥|V_{n}|^{1/2}(−Δ + 1)^{−1/2}Q_{K}∥ = 0  (4.16)

Thus, by a standard diagonalization argument, it suffices to show that for each K, there is a subsequence so that

lim_{j,k→∞} ∥(−Δ + 1)^{−1/2}P_{K}(V_{n(j)} − V_{n(k)})P_{K}(−Δ + 1)^{−1/2}∥ = 0 (4.17)

In momentum space,

Q_{n} = (−Δ + 1)^{−1/2}P_{K}V_{n}P_{K}(−Δ + 1)^{−1/2}  (4.18)

has an integral kernel

Q_{n}(p, q) = χ_{|p|≤K}(p)(p^{2} + 1)^{−1/2}V̂_{n}(p − q)χ_{|q|≤K}(q)(q^{2} + 1)^{1/2} (4.19)

By (4.12) and the fixed support hypothesis, we have

sup_{n} (∥V_{n}∥_{L_{1}} + ∥x⃗ V_{n}∥_{L^{1}}) < ∞  (4.20)

so that

sup_{n} (|V̂_{n}(k)| + |∇V̂_{n}(k)|) < ∞  (4.21)

which means {V_{n}(k) | |k| ≤ 2K} is a uniformly equicontinuous family, so we can find a subsequence so

lim_{j,k→∞} sup_{|k|≤2K} |V̂_{n(j)}(k) − V̂_{n(ℓ)}(k)| = 0  (4.22)

It follows from (4.19) that

∫ |Q_{n(j)}(p, q) − Q_{n(ℓ)}(p, q)|^{2} dpdq → 0  (4.23)

so (4.17) holds since the Hilbert-Schmidt norm dominates the operator norm. □

Given V ∈ K_{ν}, we say H̃ is a limit point at infinity in direction e if there exists x_{n} → ∞ with x_{n}/|x_{n}| → e so that for the characteristic function of any ball and z ∈ ℂ\[a, ∞), we have

lim_{n→∞} ∥[(−Δ + V (x − x_{n}) − z)^{−1} − (H̃ − z)^{−1}]χ_{S}∥ = 0  (4.24)

Let L_{e} denote the set of limit points in direction e. Then our standard argument using Theorem 4.2 and Proposition 4.4 to get compactness implies

Theorem 4.5. Let V ∈ K_{ν}. Then

σ_{ess}(−Δ + V) = \overline{⋃_{e} ⋃_{H̃∈L_{e}} σ(H̃)}  (4.25)

5. Potentials Asymptotic to Isospectral Tori

As a warmup, we will prove the following result which includes Theorem 1.2 as a special case. We will consider functions f: ℝ^{ν} → ℝ^{ν} so

lim_{|x|→∞} sup_{|y|≤L} |f(x) − f(x + y)| = 0  (5.1)

for each L. For example, if f is C^{1} outside some ball and |∇f(x)| → 0 (e.g., f(x) = \sqrt{x}\frac{x}{|x|}), then (5.1) holds.

Theorem 5.1. Let V be a function on ℝ^{ν}, periodic in ν independent directions, so V is uniformly Kato (e.g., V ∈ L_{loc}^{p} with p a canonical value for ℝ^{ν}). Suppose either V is bounded or |f(x) − f(y)| ≤ (1 − ε)|x − y| for some ε > 0. Let f obey (5.1). Let W (x) = V (x + f(x)). Then

σ_{ess}(−Δ + W) = σ(−Δ + V)  (5.2)

Remark. The condition that V is bounded or f is globally Lifschitz is needed to assure W is locally L^{1}. We thank V. Georgescu for pointing out to us the need for this condition, which was missing in our original preprint.

Proof. Let L be the integral lattice generated by some set of periods so V (x + ℓ) = V (x) if ℓ ∈ L. Let π: ℝ^{ν} → ℝ^{ν}/L be the canonical projection. If x_{j} ∈ ℝ^{ν}, since ℝ^{ν}/L is compact, we can find a subsequence m(j) so π((x_{m(j)}) + f(x_{m(j)})) → x_{∞}. Then

−Δ + W (· − x_{m(j)}) → −Δ + V (x − x_{∞})

so the limits are translates of −Δ+V, which all have the same essential spectrum. (5.2) is immediate from Theorem 3.12. □

Our next result includes Theorem 1.3.

Theorem 5.2. Let W: ℝ^{d} → ℝ be bounded and continuous, and obey

W (x + a) = W (x)  (5.3)

if a ∈ ℤ^{d}. Let (α_{1}, …, α_{d}) be such that {(α_{1}n, α_{2}n, …, α_{d}n) | n ∈ ℤ} is dense in ℝ^{d}/ℤ^{d} (i.e., 1, α_{1}, …, α_{d} are rationally independent). Let f: ℤ → ℝ^{d} obey

lim_{n→∞} sup_{|m|≤L} |f(n) − f(n + m)| = 0

for each L. Let V_{0}(n) = W (αn) and let

V (n) = W (αn + f(n))  (5.4)

On ℓ^{2}(ℤ), let (h_{0}u)(n) = u(n + 1) + u(n − 1). Then

σ_{ess}(h_{0} + V) = σ(h_{0} + V_{0})  (5.5)

Proof. For each x ∈ ℝ^{d}/ℤ^{d}, define

V_{x}(n) = W (αn + x)  (5.6)

Then a theorem of Avron-Simon [7] (see [19]) shows that σ(h_{0} + V_{x}) is independent of x (and purely essential). Given any sequence n_{j}, find a sequence n_{j(m)} so f(n_{j(m)}) → x_{∞} in ℝ^{d}/ℤ^{d}. Then V (n + n_{j(m)}) → V_{x_{∞}}(n), so by Theorem 1.7,

σ_{ess}(h_{0} + V) = \overline{⋃_{x} σ(h_{0} + V_{x}) = σ(h_{0} + V_{0})}  □

Next, we turn to Theorem 1.5 in the OPUC case. Any set of periodic Verblunsky coefficients {α_{n}}_{n=0}^{∞} with

α_{n+p} = α_{n}  (5.7)

for some p defines a natural function on ℂ\{0}, Δ(z) = z^{−p/2}Tr(T_{p}(z)), where T_{p} is a transfer matrix; see Section 11.1 of [77]. (If p is odd, Δ is double-valued; see Chapter 11 of [77] for how to handle odd p.) Δ is real on ∂𝔻 and σ_{ess}(C(α)) is a union of ℓ disjoint intervals; ℓ ≤ p (generically, ℓ = p). As proven in Chapter 11 of [77],

{β ∈ 𝔻^{p} | Δ(z; {β_{n mod p}}_{n=0}^{∞}) = Δ(z; α)} ≡ T_{α}  (5.8)

is an ℓ-dimensional torus called the isospectral torus. Moreover, the two-sided CMV matrix, defined by requiring (5.8) for all n ∈ ℤ, has

σ(C̃(β)) = σ_{ess}(C(α))  (5.9)

for any β ∈ T_{α}.

Given two sequences {κ_{n}}_{n=0}^{∞} and {λ_{n}}_{n=0}^{∞} in 𝔻^{p}, define

d(κ, λ) ≡ ∑_{n=0}^{∞} e^{−n}|κ_{n} − λ_{n}|  (5.10)

Convergence in d-norm is the same as sequential convergence. We define

d(κ, T_{α}) = inf_{β∈T_{α}} d(κ, β)

A sequence γ_{n} is called asymptotic to T_{α} if

lim_{m→∞} d({γ_{n+m}}_{n=0}^{∞}, T_{α}) = 0  (5.11)

Then the OPUC case of Theorem 1.5 (settling Conjecture 12.2.3 of [77]) says

Theorem 5.3. Let (5.11) hold. Then

σ_{ess}(C({γ_{n}}_{n=0}^{∞})) = σ_{ess}(C({α_{n}}_{n=0}^{∞}))  (5.12)

Proof. The right limit points are a subset of {C̃({β_{n mod p}}_{n=−∞}^{∞}) | {β}_{n=0}^{p−1} ∈ T_{α}}, so by Theorem 3.1 and (5.9), (5.12) holds. □

By the same argument using isospectral tori for periodic Jacobi matrices [23, 45, 46, 82] and for Schrödinger operators [21, 51, 58], one has

Theorem 5.4. If T is the isospectral torus of a given periodic Jacobi matrix, J̃, and J has Jacobi parameters obeying

lim_{n→∞} min_{ã,b̃∈T} ∑_{ℓ=1}^{∞} [|a_{n+ℓ} − ã_{ℓ}| + |b_{n+ℓ} − b̃_{ℓ}|]e^{−ℓ} = 0  (5.13)

then

σ_{ess}(J) = σ(J̃)  (5.14)

Theorem 5.5. Let T be the isospectral torus of a periodic potential, V_{0}, on ℝ and V on [0, ∞) in K_{1} and

lim_{|x|→∞} inf_{W ∈T} ∫_{0}^{∞}|V (y + x) − W (y)|e^{−|y|} dy = 0  (5.15)

then

σ_{ess}(−\frac{d^{2}}{dx^{2}} + V ) = σ(−\frac{d^{2}}{dx^{2}} + V_{0})  (5.16)

where −\frac{d^{2}}{dx^{2}} + V is defined on L^{2}(0, ∞) with u(0) = 0 boundary conditions and −\frac{d^{2}}{dx^{2}} + V_{0} is defined on L^{2}(ℝ, dx).

The following provides an alternate proof of Theorem 4.3.8 of [76]:

Theorem 5.6. Let {α_{j}}_{j=0}^{∞} and {β_{j}}_{j=0}^{∞} be two sequences of Verblunsky coefficients. Suppose there exist λ_{j} ∈ ∂𝔻 so that

(i)  β_{j}λ_{j} − α_{j} → 0  (5.17)
(ii)  λ_{j+1}λ̄_{j} → 1  (5.18)

Then

σ_{ess}(C({α_{j}}_{j=0}^{∞})) = σ_{ess}(C({β_{j}}_{j=0}^{∞}))  (5.19)

Proof. Let {γ_{j}}_{j=−∞}^{∞} be a right limit of {β_{j}}_{j=0}^{∞}, that is, β_{ℓ+n_{k}} → γ_{ℓ} for some n_{k}. By passing to a subsequence, we can suppose λ_{n_{j}} → λ_{∞}, in which case (5.18) implies λ_{n_{j}+ℓ} → λ_{∞} for each ℓ fixed. By (5.17), {λ_{∞}γ_{j}}_{j=−∞}^{∞} is a right limit of {α_{j}}_{j=0}^{∞}. Since σ(C̃({λγ_{j}}_{j=−∞}^{∞})) is λindependent, (5.19) follows from (3.15). □

6. The HVZ Theorem

For simplicity of exposition, we begin with a case with an infinityheavy particle; eventually we will consider a situation even more general than arbitrary N-body systems. Thus, H acts on L^{2}(ℝ^{μ(N−1)}, dx) with

H = − ∑_{j=1}^{N−1} (2m_{j})^{−1}Δ_{x_{j}} + ∑_{j=1}^{N−1} V_{0j}(x_{j}) + ∑_{1≤i<j≤N−1} V_{ij}(x_{j} − x_{i}) (6.1)

where x = (x_{1}, …, x_{N−1}) with x_{j} ∈ ℝ^{μ}. Here the V ’s will be in K_{μ} with K_{μ} vanishing at infinity. a will denote a partition (C_{1} … C_{ℓ}) of {0, …, N − 1} onto ℓ ≥ 2 clusters. We say (ij) ⊂ a if i, j are in the same cluster, C ∈ a, and (ij) ⊄ a if i ∈ C_{k} and j ∈ C_{m} with k ≠ m,

H(a) = H − ∑_{\begin{matrix}ij⊄a \\ i<j \end{matrix}} V_{ij}(x_{j} − x_{i})  (6.2)

with x_{0} ≡ 0. The HVZ theorem says that Theorem 6.1. If each V_{ij} is in K_{μ}, K_{μ} vanishing at infinity, then

σ_{ess}(H) = \overline{⋃_{a} σ(H(a))}  (6.3)

Since H(a) commutes with translations of clusters, H has the form H(a) = T^{a} ⊗ 1 + 1 ⊗ H_{a} where T^{a} is a Laplacian on ℝ^{μ(ℓ−1)}, and thus, if Σ(a) = inf σ(H_{a}), then σ(H(a)) = [Σ(a), ∞). So (6.3) says

σ_{ess}(H) = [Σ, ∞) Σ ≡ inf_{a} Σ(a)  (6.4)

This result is, of course, well-known, going back to Hunziker [36], van Winter [83], and Zhislin [87], with geometric proofs by Enss [22], Simon [74], Sigal [73], and Garding [24]. Until Garding [24], all proofs involved some kind of combinatorial argument if only the existence of a Ruelle-Simon partition of unity. Like Garding [24], we will be totally geometric with a straightforward proof exploiting our general machine. C^{∗}-algebra proofs can be found in Georgescu-Iftimovici [26, 27] and have a spirit close to our proof below. Rabinovich [66] has a proof of HVZ using his notion of invertibility at infinity that also has overlap with our philosophy.

There is one subtlety to mention. Consider the case μ = 1, N = 3, so ℝ^{μ(N−1)} = ℝ^{2} = {(x_{1}, x_{2}) | x_{1}, x_{2} ∈ ℝ}. There are then clearly six special directions: ±(1, 0), ±(0, 1), and ±(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}). For any other direction ê, if x_{n}/|x_{n}| → ê, V → 0, and the limit in that direction is H_{0} = H({0}, {1}, {2}).

For e = ±(1, 0), |(x_{n})_{1}| → ∞ and |(x_{n})_{1} − (x_{n})_{2}| → ∞, so the only limit at infinity would appear to be H({0, 2}, {1}). But this is wrong! To say x_{n} has limit ±(1, 0) says x_{n}/|x_{n}| → ±(1, 0), so (x_{n})_{1} → ±∞. But it does not say (x_{n})_{2} → 0, only (x_{n})_{2}/(x_{n})_{1} → 0. For example, if (x_{n})_{2} → ∞, the limit is H_{0}. As we will see (it is obvious!), the limits are precisely H_{0} and translates of H({0, 2}, {1}). This still proves (6.3), but with a tiny bit of extra thought needed.

We want to note a general form for extending HVZ due to Agmon [1]. We consider linear surjections π_{j}: ℝ^{ν} → ℝ^{μ_{j}} with μ_{j} ≤ ν. Let V_{j}: ℝ^{μ_{j}} → ℝ be in K_{μ_{j}} vanishing in K_{μ_{j}} sense at infinity. Then

H = −Δ + ∑_{j} V_{j}(π_{j}x)  (6.5)

will be called an Agmon Hamiltonian.

Given e ∈ S^{ν−1}, define

H_{e} = −Δ + ∑_{{j|π_{j}e=0}} V_{j}(π_{j}x) ≡ −Δ + V_{e}  (6.6)

Notice that since H_{e} commutes with x → x+λe, H_{e} has the form H_{e} = −Δ_{e} ⊗1+1⊗(−Δ_{e^{⊥}} +V_{e}), so σ(H_{e}) = [Σ_{e}, ∞) with Σ_{e} = inf spec(H_{e}).

In general, if ∩_{j} ker(π_{j}) ≠ {0}, H has some translation invariant degrees of freedom and can, and should, be reduced, but the HVZ theorem holds for the unreduced case (and also for the reduced case, since the reduced H which acts on ℝ^{ν}/ ∩_{j} ker(π_{j}) has the form (6.5)). So we will not consider reduction in detail.

By using π_{j} to write V_{ij}(x_{i} − x_{j}) in terms of mass scaled reduced coordinates, any N-body Hamiltonian has the form (6.5), and (6.5) allows many-body forces. For the case of Theorem 6.1, if e is given, define a to be the partition with (ij) ⊂ a if and only if e_{i} = e_{j} (with e_{0} ≡ 0). Then H_{e} = H(a) and (6.7) below is (6.3).

Theorem 6.2. For any Agmon Hamiltonian,

σ_{ess}(H) = \overline{⋃_{e∈S^{ν−1}} σ(H_{e})}  (6.7)

Proof. If x_{n}/|x_{n}| → e, we can pass to a subsequence where each π_{j}x_{n} has a finite limit, or else has |π_{j}x_{n}| → ∞. It follows that the limit at infinity for x_{n} is a translation (by lim π_{j}x_{n}) of H_{e} or of a limit at infinity of H_{e}. Thus, for any H̃ in L_{e}, the set of limits in direction e,

σ(H̃) ⊂ σ(H_{e})

and so,

\overline{⋃_{H̃∈L_{e}} σ(H̃}) = σ(H_{e})

and (6.7) is (4.25). □

Remark. It is not hard to see that as e runs through S^{ν−1}, σ(H_{e}) has only finitely many distinct values, so the closure in (6.7) is superfluous.

Because we control σ_{ess}(H) directly and do not rely on the a priori fact that one only has to properly locate inf σ_{ess}(H) (as do all the proofs quoted above, except the original H,V,Z proofs and Simon [74]), we can obtain results on N-body interactions where the particles move in a fixed background periodic potential with gaps that can produce gaps in σ_{ess}(H).

7. Additional Applications

We want to consider some additional applications of our machinery that shed light on earlier works:

(a) Sparse bumps, already considered by Klaus [44] using Birman-Schwinger techniques, by Cycon et al. [19] using geometric methods, and by Hundertmark-Kirsch [35] using methods that are essentially the same as the specialization of our argument to this example. Georgescu-Iftimovici [27] also have a discussion of sparse potentials that overlaps our discussion.

(b) Jacobi matrices with a_{n} → 0 and CMV matrices with |α_{n}| → 1 already studied by Maki [53], Chihara [13] (Jacobi), and by Golinskii [29] (CMV).

(c) Bounded Jacobi matrices and CMV matrices with finite essential spectrum already studied by Krein (in [3]) and Chihara [14] (Jacobi case), and by Golinskii [29] (CMV case).

Remark. Golinskii [29] for (b) and (c) did not explicitly use CMV matrices but rather studied measures on ∂𝔻, but his results are equivalent to statements about CMV matrices.

Here is the sparse potentials result:

Theorem 7.1 ([44, 19]). Let W be an L^{1} potential of compact support on ℝ. Let x_{0} < x_{1} < ⋯ < x_{n} < ⋯ so x_{n+1} − x_{n} → ∞. Let

V (x) = ∑_{j=0}^{∞} W (x − x_{j})  (7.1)

Then

σ_{ess}(−\frac{d^{2}}{dx^{2}} + V (x)) = σ(−\frac{d^{2}}{dx^{2}} + W )  (7.2)

Remarks. 1. That W has compact support is not needed. W (x) → 0 sufficiently fast (e.g., bounded by x^{−1−ε}) will do with no change in proof.

2. Discrete eigenvalues of −\frac{d^{2}}{dx^{2}} + W are limit points of eigenvalues for −\frac{d^{2}}{dx^{2}} + V.

3. There is a higher-dimensional version of this argument; see [35].

Proof. The limits at infinity are −\frac{d^{2}}{dx^{2}} and −\frac{d^{2}}{dx^{2}} + W (x − a). Now use Theorem 3.12 or Theorem 4.5. □

Remark. This example is important because it shows that one needs σ(H̃) and not just σ_{ess}(H̃).

As for a_{n} → 0:

Theorem 7.2 ([13]). Let J be a bounded Jacobi matrix with a_{n} → 0. Let S be the limit points of {b_{n}}_{n=1}^{∞}. Then

σ_{ess}(J) = S  (7.3)

Proof. The limit points at infinity are diagonal matrices with diagonal matrix elements in S, and by a compactness argument, every s ∈ S is a diagonal matrix element of some limit. Theorem 3.3 implies (7.3). □

Theorem 7.3 ([29]). Let C({α_{n}}_{n=0}^{∞}) be a CMV matrix of a sequence of Verblunsky coefficients with

lim_{n→∞} |α_{n}| = 1  (7.4)

Let S be the set of limit points of {−ᾱ_{j+1}α_{j}}. Then

σ_{ess}(C({α_{j}}_{j=1}^{∞})) = S  (7.5)

Proof. By compactness of ∂𝔻, if s ∈ S, there is a sequence n_{j} so α_{n_{j}+ℓ} has a limit, β_{ℓ}, for all ℓ and s = −β̄_{1}β_{0}. The limiting CMV matrices have |β_{ℓ}| = 1 by (7.4), so are diagonal with matrix elements −β̄_{ℓ+1}β_{ℓ}. Thus, the spectra of limits lie in S, and by the first sentence, any such s ∈ S is in the spectrum of a limit. Now use Theorem 3.2. □

Finally, we turn to the case of finite essential spectrum, first for Jacobi matrices.

Theorem 7.4. Let x_{1}, …, x_{ℓ} ∈ ℝ be distinct. A bounded Jacobi matrix J has

σ_{ess}(J) = {x_{1}, …, x_{ℓ}}  (7.6)

if and only if (i)

lim_{n→∞} a_{n}a_{n+1} … a_{n+ℓ−1} = 0  (7.7)

(ii) If k ≤ l and n_{j} is such that

a_{n_{j}} → 0  a_{n_{j}+k} → 0  (7.8)
a_{n_{j}+m} → ã_{m} ≠ 0  m = 1, 2, …, k − 1  (7.9)
b_{n_{j}+m} → b̃_{m}  m = 1, 2, …, k  (7.10)

then the finite k × k matrix,

J̃ = (\begin{matrix}b̃_{1} & ã_{1} \\ ã_{1} & b̃_{2} & ã_{2} \\ & ... & ... & ... \\ & & ... & ... & ã_{k} \\ & & & ã_{k−1} & b̃_{k} \end{matrix})  (7.11)

has spectrum a k-element subset of {x_{1}, …, x_{ℓ}}. (iii) Each x_{j} occurs in at least one limit of the form (7.11)

Proof. By Theorem 3.3, (7.6) holds if and only if the limiting J̃’s have spectrum in {x_{1}, …, x_{ℓ}} and there is at least one J̃ with each x_{j} in the spectrum. J̃ is a direct sum of finite and/or semi-infinite and/or infinite pieces. The semi-infinite pieces correspond to Jacobi matrices with nontrivial measures which have infinite spectrum. The two-sided infinite pieces also have infinite spectrum. Finite pieces of length m, which have a’s nonzero, have m points in their spectrum, so no limit can have a direct summand of length ℓ + 1 or more. Thus, by compactness, (7.7) holds, that is, any set of ℓ a’s in the limit must have at least one zero. (ii) is then the assertion that the limits have spectrum in {x_{1}, …, x_{ℓ}}, and (iii) is that each x_{j} occurs. □

Theorem 7.5. (a) J obeys

σ_{ess}(J) ⊂ {x_{1}, …, x_{ℓ}}  (7.12)

if and only if every right limit, J̃, obeys

∏_{j=1}^{ℓ}(J̃ − x_{j}) ≡ P (J̃) = 0  (7.13)

(b) J obeys (7.12) if and only if P (J) is compact.

Proof. (a) (7.13) holds if and only if σ(J̃) ⊂ {x_{1}, …, x_{ℓ}}, so this follows from Theorem 3.3.

(b) P (J) has finite width. Thus, it is compact if and only if all matrix elements go to zero, which is true (by compactness of translates of J) if and only if (7.13) holds for all limits. □

We have now come full circle—for Theorem 7.5(b) is precisely Krein’s criterion (stated in [3]), whose proof is immediate by the spectral mapping theorem and the analysis of the spectrum of compact selfadjoint operators. However, our Theorem 7.4 gives an equivalent, but subtly distinct, way to look at the limits. To see this, consider the case ℓ = 2, that is, two limiting eigenvalues x_{1} and x_{2}.

This has been computed by Chihara [15], who found necessary and sufficient conditions for σ_{ess}(J) = {x_{1}, x_{2}} are (there is a typo in [15], where we give (b_{n} − x_{1})(b_{n} − x_{2}) in (7.14); he gives, after changing to our notation, (b_{n} − x_{1})(b_{n+1} − x_{2})):

lim_{n→∞} (a_{n}^{2} + a_{n−1}^{2} + (b_{n} − x_{1})(b_{n} − x_{2})) = 0  (7.14)
lim_{n→∞} (a_{n}(b_{n} + b_{n+1} − x_{1} − x_{2})) = 0  (7.15)
lim_{n→∞} (a_{n}a_{n+1}) = 0  (7.16)

To see this from the point of view of (J − x_{1})(J − x_{2}), note that

⟨δ_{n}, (J − x_{1})(J − x_{2})δ_{n}⟩ = a_{n}^{2} + a_{n−1}^{2} + (b_{n} − x_{1})(b_{n} − x_{2}) (7.17)
⟨δ_{n+1}, (J − x_{1})(J − x_{2})δ_{n}⟩ = a_{n}(b_{n} − x_{2}) + a_{n}(b_{n+1} − x_{1})  (7.18)
⟨δ_{n+2}, (J − x_{1})(J − x_{n})δ_{n}⟩ = a_{n}a_{n+1}  (7.19)

If we think in terms of limit points, we get a different-looking set of equations. Consider limits, J̃. Of course, (7.16) is common

ã_{n}ã_{n+1} = 0  (7.20)

But the conditions on summands of J̃ become

ã_{n} = ã_{n−1} = 0 ⇒ b̃_{n} = x_{1} or b̃_{n} = x_{2}  (7.21)
ã_{n} ≠ 0 ⇒ b̃_{n+1} + b̃_{n} = x_{1} + x_{2} and b̃_{n}b̃_{n+1} − ã_{n}^{2} = x_{1}x_{2} (7.22)

For (7.21) is the result for 1 × 1 blocks, and (7.22) says 2 × 2 blocks have eigenvalues x_{1} and x_{2}. It is an interesting exercise to see that (7.20)–(7.22) are equivalent to

ã_{n}^{2} + ã_{n+1}^{2} + (b̃_{n} − x_{1})(b̃_{n} − x_{2}) = 0  (7.23)
ã_{n}(b̃_{n} + b̃_{n+1} − x_{1} − x_{2}) = 0  (7.24)
ã_{n}ã_{n+1} = 0  (7.25)

One can analyze CMV matrices similar to the above analysis. The analog of Theorem 7.4 is:

Theorem 7.6. Let λ_{1}, …, λ_{ℓ} ∈ ∂𝔻 be distinct. A CMV matrix C has

σ_{ess}(C) = {λ_{1}, …, λ_{ℓ}}  (7.26)

if and only if (i)

lim_{n→∞} ρ_{n}ρ_{n+1} … ρ_{n+ℓ−1} = 0  (7.27)

(ii) If k ≤ ℓ and n_{j} is such that

ρ_{n_{j}} → 0  ρ_{n_{j}+k} → 0  (7.28)
α_{n_{j}+m} → α̃_{m}  m = 0, 1, 2, …, k − 1, k

with |α̃_{m}| ≠ 1, m = 1, …, k − 1 (by (7.28), |α̃_{0}| = |α̃_{k}| = 1), then the matrix (1 = 1 × 1 unit matrix)

C̃ = [Θ(α̃_{1})⊕⋯⊕Θ(α̃_{k−1})][−α̃_{0}1⊕Θ(α̃_{2})⊕⋯⊕Θ(α̃_{k−2})⊕ ¯α̃_{k}1] (7.29)

if k is even and

C̃ = [Θ(α̃_{1})⊕⋯⊕Θ(α̃_{k−2})⊕ ¯α̃_{k}1][−α̃_{0}1⊕Θ(α̃_{2})⊕⋯⊕Θ(α̃_{k−1})] (7.30)

if k is odd has eigenvalues k elements among λ_{1}, …, λ_{ℓ}.

(iii) Each of λ_{1}, …, λ_{ℓ} occurs as an eigenvalue of some C̃.

Proof. Same as Theorem 7.4. □

The analog of Theorem 7.5 is

Theorem 7.7. Let λ_{1}, …, λ_{ℓ} ∈ ∂𝔻 be distinct.

(a) C obeys

σ_{ess}(C) ⊂ {λ_{j}, …, λ_{ℓ}}  (7.31)

if and only if every right limit C̃ obeys

∏_{j=1}^{ℓ}(C̃ − λ_{j}) ≡ P (C̃) = 0  (7.32)

(b) C obeys (7.31) if and only if P (C) is compact.

Proof. Same as Theorem 7.5. □

We have come to Golinskii’s OPUC analog of Krein’s theorem [29]. Again, it is illuminating to consider the case ℓ = 2. We will deal directly with limits of α_{j}, call them α̃_{j}. The Theorem 7.6 view of things is

ρ̃_{n}ρ̃_{n+1} = 0  (7.33)
ρ̃_{n} = ρ̃_{n+1} = 0 ⇒ −¯α̃_{n+1}α̃_{n} = λ_{1} or − ¯ α̃_{n+1}α̃_{n} = λ_{2}  (7.34)

ρ̃_{n} ≠ 0 ⇒ −¯α̃_{n}α̃_{n−1} − ¯ α̃_{n+1}α̃_{n} = λ_{1} + λ_{2} and α̃_{n−1} ¯α̃_{n+1} = λ_{1}λ_{2} (7.35)

(7.35) comes from the fact that the matrix C of (7.29) is

(\begin{matrix}¯α̃_{n} & ρ̃_{n} \\ ρ_{n} & −α̃_{n} \end{matrix}) (\begin{matrix}−α̃_{n−1} & 0 \\ 0 & ¯α̃_{n+1} \end{matrix})  (7.36)

where the determinant is α̃_{n−1} ¯α̃_{n+1} and the trace is −¯α̃_{n}α̃_{n−1} −α̃_{n} ¯α̃_{n+1}.

From the point of view of Theorem 7.7, using the CMV matrix is complicated since (C − λ_{1})(C − λ_{2}) is, in general, 9-diagonal! As noted by Golinskii [29], it is easier to use the GGT matrix (see Section 4.1 of [76]), since it immediately implies

ρ̃_{n}ρ̃_{n+1} = ⟨δ_{n+2}, G^{2}δ_{n}⟩ = 0  (7.37)

and once that holds, G becomes tridiagonal! Thus, one gets from ⟨δ_{n+1}, (G − λ_{1})(G − λ_{2})δ_{n}⟩ = 0 that

ρ̃_{n}(−¯α̃_{n}α̃_{n−1} − ¯ α̃_{n+1}α̃_{n} − λ_{1} − λ_{2}) = 0  (7.38)

and from ⟨δ_{n}, (G − λ_{1})(G − λ_{2})δ_{n}⟩ = 0,

(−¯α̃_{n+1}α̃_{n} − λ_{1})(−¯α̃_{n}α̃_{n−1} − λ_{2}) − ρ̃_{n}^{2} ¯α̃_{n+1}α̃_{n−1} − ρ_{n+1}^{2} ¯α̃_{n−2}α̃_{n+1} = 0 (7.39)

Again, it is an interesting exercise that (7.33)–(7.35) are equivalent to (7.37)–(7.39).

8. Magnetic Fields

A magnetic Hamiltonian acts on ℝ^{ν} via

H(a, V) = − ∑_{j=1}^{ν} (∂_{j} − ia_{j})^{2} + V  (8.1)

where a is vector-valued. The magnetic field is the two-form defined by

B_{jk} = ∂_{j}a_{k} − ∂_{k}a_{j}  (8.2)

If λ is a scalar function, then

ã = a + ∇λ  (8.3)

produces the same B, and one has gauge covariance

H(ã, V) = e^{iλ}H(a, V)e^{−iλ}  (8.4)

While the mathematically “natural” conditions on a are either a ∈ L_{loc}^{4}, ∇ · a ∈ L_{loc}^{2}, or a ∈ L_{loc}^{2} (see [19, 50, 75]), for simplicity, we will suppose here that B is bounded and uniformly Hölder continuous, that is, for some δ > 0,

sup_{x,j,k} |B_{jk}(x)| < ∞  sup_{j,k,|x−y|≤1} |x − y|^{−δ}|B_{jk}(x) − B_{jk}(y)| < ∞ (8.5)

It is certainly true that one can allow suitable local singularities. We will see later what (8.5) implies about choices of a. With this kind of regularity on B, it is easy to prove that for a shift between different gauges of the type we consider below, the formal gauge covariance (8.4) is mathematically valid. Indeed, more singular gauge changes can be justified (see Leinfelder [49]).

If a_{j} → 0 at infinity, it is easy to implement the ideas of Sections 3 and 4 with no change in the meaning of limit point at infinity; the limits all have no magnetic field. But as is well known, a_{j} → 0 requires, very roughly speaking, that B goes to zero at least as fast as |x|^{−1−ε}, so this does not even capture all situations where B_{ij} → 0 at infinity. Miller [59] (see also [19, 60]) noted that, in two and three dimensions, the way to control B → 0 at infinity is to make suitable gauge changes in Weyl sequences—and that will also be the key to what we do here.

We will settle for stating a very general limit theorem and not attempt to apply this theorem to recover the rather extensive literature on HVZ theorems and on essential spectra in periodic magnetic fields [5, 10, 18, 31, 32, 33, 34, 37, 38, 41, 62, 63, 81, 84, 85, 86, 88, 89, 90, 91].

We have no doubt that can be done and that the ideas below will be useful in future studies. We note that it should be possible to extend Theorem 5.1 with “slipped periodic” magnetic fields.

Definition. A set of gauges, a_{x}, depending on x is said to be “regular at infinity” if and only if, for every R, we have for some δ > 0,

sup_{|x−y|≤R} |a_{x}(y)| < ∞  sup_{\begin{matrix}x,y,z \\ |y−z|<1 \\ |x−y|<R \end{matrix}} |y − z|^{−δ}|a_{x}(y) − a_{x}(z)| < ∞ (8.6)

Proposition 8.1. If (8.5) holds, there exists a set of gauges regular at infinity.

Proof. The transverse gauge, a⃗_{x_{0}}, based at x_{0} is defined by

a_{x_{0};j}(x_{0} + y) = ∑_{k}[∫_{0}^{1} sB_{kj}(x_{0} + sy) ds]y_{k}  (8.7)

That this is a gauge is known (see below), and clearly, if |x_{0} − y| ≤ R,

|a⃗_{x_{0}}(y)| ≤ \frac{1}{2} R sup_{x} ∥B(x)∥

and if |y − z| < 1 and |x_{0} − y| < R,

|a⃗_{x_{0}}(y)−a⃗_{x_{0}}(z)| ≤ \frac{1}{2} \{sup_{x} ∥B(x)∥+(R−1) sup_{|y−z|≤1} [|y−z|^{−δ}∥B(y)−B(z)∥]\}

□

Remarks. 1. We will call the choice (8.7) the local transverse gauge.

2. Transverse gauge goes back at least to Uhlenbeck [80], who calls them exponential gauge. They have been used extensively by Loss-Thaller [52] (see also Thaller [79]) to study scattering.

3. To see that (8.7) is a gauge is a messy calculation if done directly, but there is a lovely indirect argument of Uhlenbeck [80]. Without loss, take x_{0} = 0. Call a gauge transverse if a⃗(0) = 0 and x⃗ · a⃗ = 0. Transverse gauges exist, for if a⃗_{0} is any gauge and

φ(x⃗) = − ∫_{0}^{1} x⃗ · a_{0}(sx⃗) ds  (8.8)

then x⃗ · ∇φ = r\frac{∂}{∂r}φ = −x⃗ · a_{0}(x), so a = a_{0} + ∇φ is transverse. Next, note that if a⃗ is a transverse gauge, then

∑ x_{k}B_{kj} = (x ·∇)a_{j} − ∇_{j}(x · a) + a_{j}
= \frac{∂}{∂r} ra_{j}  (8.9)

Integrating (8.9) shows (8.7) with y = 0 is not only a gauge but the unique transverse gauge.

If a_{x} is a set of gauges regular at infinity, we say H̃ is a limit at infinity of H(a, V) in direction ê if and only if with

(U_{x}φ)(y) = φ(y − x)  (8.10)

we have that for some sequence x_{n}, |x_{n}| → ∞, x_{n}/|x_{n}| → e, and for each R < ∞ and z ∈ ℂ\[α, ∞),

U_{x_{n}}((H(a_{x_{n}}, V) − z)^{−1})U_{x_{n}}^{−1}χ_{R} → (H̃ − z)^{−1}χ_{R}  (8.11)

with χ_{R} the characteristic function of a ball of radius R about 0. As usual, L_{e} denotes the limits at infinity in direction e.

Theorem 8.2. If V ∈ K_{ν} and B obeys (8.5), then

σ_{ess}(H(a, V)) = \overline{⋃_{e∈S^{ν−1}} ⋃_{H̃∈L_{e}} σ(H̃)}  (8.12)

In (8.12), we get the same union if, instead of all regular gauges at infinity, we take only the local transverse gauges.

Proof. By using gauge-transformed Weyl sequences as in [19], it is easy to see the right side of (8.12) is contained in σ_{ess}(H(a, V)). To complete the proof, we need only show the right side, restricted to local transverse gauges, contains σ_{ess}(H(a, V)).

Localization extends effortlessly since [j, H(a, V)] = ∇j · (∇ − ia⃗) + (∇−ia⃗) · ∇j and ∥(∇−ia⃗)φ∥^{2} is controlled by H(a, V). Thus, we only need compactness of the gauge-transformed operators. Since (8.6) says the a_{x}’s translated to 0 are uniformly equicontinuous, compactness of the a’s is immediate. V ’s are handled as in Section 4. □

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