VARIATIONAL ESTIMATES FOR

DISCRETE SCHRÖDINGER OPERATORS WITH POTENTIALS OF INDEFINITE SIGN

D. DAMANIK^{1,3}, D. HUNDERTMARK^{2}, R. KILLIP^{1}, AND B. SIMON^{1,4}

Abstract. Let H be a one-dimensional discrete Schrödinger operator. We prove that if σ_{ess}(H) ⊂ [−2, 2], then H −H_{0} is compact and σ_{ess}(H) = [−2, 2]. We also prove that if H_{0} + \frac{1}{4} V^{2} has at least one bound state, then the same is true for H_{0} + V. Further, if H_{0} + \frac{1}{4} V^{2} has infinitely many bound states, then so does H_{0} + V. Consequences include the fact that for decaying potential V with lim inf_{|n|→∞} |nV (n)| > 1, H_{0} + V has infinitely many bound states; the signs of V are irrelevant. Higher-dimensional analogues are also discussed.

1. Introduction

Let H be a Schrödinger operator on ℓ^{2}(ℤ),

(Hu)(n) = u(n + 1) + u(n − 1) + V (n)u(n)  (1.1)

with bounded potential V : ℤ → ℝ. The free Schrödinger operator, H_{0}, corresponds to the case V = 0. One of our main results in this paper is

Theorem 1. If σ_{ess}(H) ⊂ [−2, 2], then V (n) → 0 as |n| → ∞, that is, H − H_{0} is compact.

Remark. By Weyl’s Theorem, we have the immediate corollary that σ_{ess}(H) = [−2, 2] if and only if V (n) → 0.

Our motivation for this result came from two sources:

Theorem 2 (Killip-Simon [7]). If σ(H) ⊂ [−2, 2], then V = 0.

Theorem 3 (Rakhmanov [12]; see also Denisov [5], Nevai [11], and references therein). Let J be a general half-line Jacobi matrix on ℓ^{2}(ℤ^{+}),

(Ju)(n) = a_{n}u(n + 1) + b_{n}u(n) + a_{n−1}u(n − 1)  (1.2)

where a_{n} > 0 and ℤ_{+} = {1, 2, … }. Suppose that [−2, 2] is the essential support of the a.c. part of the spectral measure and also the essential spectrum. Then lim_{n→∞}|a_{n} − 1| + |b_{n}| = 0, that is, J is a compact perturbation of J_{0}, the Jacobi matrix with a_{n} ≡ 1, b_{n} ≡ 0.

Date: November 5, 2002. 1 Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125. E-mail:

damanik@its.caltech.edu; killip@its.caltech.edu; bsimon@caltech.edu.

2 Institut Mittag-Leffler, Auravägen 17, S-182 60 Djursholm, Sweden. On leave from Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801-2975.

3 Supported in part by NSF grant DMS-0227289. 4 Supported in part by NSF grant DMS-0140592.

While Theorem 3 motivated our thoughts, it is not closely related to the result. Not only are the methods different, but it holds for any a priori a_{n}; whereas our results require some a priori estimates like a_{n} → 1 as |n| → ∞. For example, if a_{n} ≡ \frac{1}{2} and b_{n} takes values +1 and −1 over longer and longer intervals, it is not hard to see that σ(J) = [−2, 2], but clearly, J −J_{0} is not compact. Thus Theorem 1, unlike Theorem 3, is essentially restricted to discrete Schrödinger operators.

For continuum Schrödinger operators, consideration of sparse positive nondecaying potentials shows that σ(H) = [0, ∞) is possible even when (H+1)^{−1}−(H_{0}+1)^{−1} is not compact. The reason is that our proof depends essentially—as does Theorem 2—on the fact that σ(H) has two sides in the discrete case.

Theorem 1 has an interesting corollary:

Corollary 4. Let H be an arbitrary one-dimensional discrete Schrödinger operator. Then sup σ_{ess}(H) − inf σ_{ess}(H) ≥ 4 with equality if and only if V (n) → V_{∞} a constant as |n| → ∞.

Proof. Let a_{+} = sup σ_{ess}(H), a_{−} = inf σ_{ess}(H). If a_{+}−a_{−} ≤ 4, then H− \frac{1}{2}(a_{+}+a_{−}) is a Schrödinger operator with essential spectrum in [−2, 2]. So Theorem 1 implies the original V (n) → \frac{1}{2}(a_{+} + a_{−}). Hence, a_{+} − a_{−} = 4 and σ_{ess} = [a_{−}, a_{+}]. □

Remarks. (a) A similar argument combined with Theorem 2 implies that if sup σ(H) − inf σ(H) ≤ 4, then V is a constant.

(b) If V (n) = (−1)^{n}λ and λ is large, standard Floquet theorem arguments show that σ(H) has two bands centered about ±(λ + O(\frac{1}{λ})) and of width O(\frac{1}{λ}). Thus, while the size of the convex hull of σ(H) is of size at least 4, the size of σ(H) can be arbitrarily small. Indeed, by results of Deift-Simon [4], if H has purely a.c. spectrum, (e.g., V periodic), the total size of σ(H) is at most 4.

While Theorem 1 is our main motivating result, the ideas behind it yield many other results about the absence of eigenvalues and about the finiteness or infinitude of their number for Schrödinger operators not only on the line, but also on the half-line or in higher dimensions. Included in our results are

(i) Theorem 1 holds in two dimensions and is false in three or more dimensions (see Theorems 4.1 and 4.2). This is connected to the fact that Schrödinger operators in one and two dimensions always have a bound state for nontrivial attractive potentials (see [9, pp. 156–157] and [8, 15]), whereas in three and more dimensions, small attractive potentials need not have bound states by the Cwikel-Lieb-Rozenblum bound [1, 10, 14].

(ii) For a half-line discrete Schrödinger operator, H, if σ(H) = [−2, 2] (i.e., no bound states), then (see Theorem 5.2)

|V (n)| ≤ 2n^{−1/2}  (1.3)

On the other hand (see Theorem 5.2), there are examples, V_{k}(n), with no bound states and lim_{k} sup_{n} n^{1/2}|V_{k}(n)| = 1. This shows that the power \frac{1}{2} in (1.3) cannot be made larger. It also shows that the constant, 2, cannot be made smaller than 1. (The optimal constant is \sqrt{2}. This is proved in [3].)

(iii) The examples in (ii) are necessarily sparse in that if |V (n)| ≥ Cn^{−α} and H has only finitely many bound states, then α ≥ 1. Indeed, we will prove (see Theorem 5.6) that if α = 1 and C > 1 or α < 1 and C > 0, then H has an infinity of bound states. This will follow from the very general theorem:

Theorem 5. Let V (n) → 0. If H_{0} + \frac{1}{4}V^{2} has at least one (resp., infinitely many) eigenvalues outside [−2, 2], then H_{0} + V has at least one (resp., infinitely many) eigenvalues outside [−2, 2].

Theorem 3.1 extends this result to all dimensions. (iv) If |V (n)| ≥ Cn^{−α} and α < 1, we will prove suitable eigenvalue moments

diverge.

The starting point of the present paper is the discussion at the end of Section 10 of [7] that it should be possible to prove Theorem 2 variationally with suitable second-order perturbation trial functions. Second-order eigenvalue perturbation theory has a change of the first-order eigenfunction by a term proportional to V. Thus, our variational trial function will have two pieces: φ and an extra piece, proportional to V φ.

The second key idea is to make use of the fact that the spectrum of H_{0} has two sides, and we can use a pair of trial functions: one to get an eigenvalue below −2 and one to get an eigenvalue above +2. By combining them, we will have various cancellations that involve terms whose sign is uncertain. Explicitly, given a pair of trial vectors φ_{+} and φ_{−}, we define

Δ(φ_{+}, φ_{−}; V) = ⟨φ_{+}, (H − 2)φ_{+}⟩ + ⟨φ_{−}, (−H − 2)φ_{−}⟩  (1.4)

where H is given by (1.1). If Δ > 0, either ⟨φ_{+}, (H − 2)φ_{+}⟩ > 0 or ⟨φ_{−}, (H + 2)φ_{−}⟩ < 0, that is, there is either an eigenvalue above 2 or below −2!

In choosing φ_{−} relative to φ_{+}, it will help to use the unitary operator U on ℓ^{2}(ℤ) given by

(Uφ)(n) = (−1)^{n}φ(n)  (1.5)

so that

UH_{0}U^{−1} = −H_{0}  UV U^{−1} = V  (1.6)

The key calculation in Section 2 will be that

Δ(φ + \frac{1}{4} V φ, U(φ − \frac{1}{4} V φ)) ≥ 2⟨φ, [H_{0} + \frac{1}{4} V^{2} − 2]φ⟩  (1.7)

For example, this immediately implies the “at least one bound state” part of Theorem 5. If H_{0} + \frac{1}{4}V^{2} has a bound state, φ, we must have ⟨φ, (H_{0} + \frac{1}{4}V^{2})φ⟩ > 2⟨φ, φ⟩, so Δ > 0.

The current paper complements [2]. That paper provided upper bounds on the distance from [−2, 2] of eigenvalues of discrete Schrödinger operators with oscillatory potentials. This paper provides lower bounds. In particular, there it was shown the Jacobi matrix with a_{n} ≡ 1, b_{n} = \frac{β(−1)^{n}}{n} has finitely many eigenvalues if |β| ≤ \frac{1}{2}. Here, we prove infinitely many (see Theorem 5.7) if |β| > 1. We also show, by ad hoc methods, that there are no eigenvalues for |β| ≤ 1 (see Proposition 5.9).

In Section 2, we prove variational estimates, including (1.7). In Section 3, we prove Theorem 5. In Section 4, we prove Theorem 1 and provide a new proof of Theorem 2. Sections 2–4 also discuss higher dimensions. In Section 5, we study the one-dimensional situation more closely.

We thank Andrej Zlatoš for useful discussions.

2. Variational Estimates

On ℓ^{2}(ℤ^{ν}), define H_{0} by

(H_{0}u)(n) = ∑_{|j|=1} u(n + j)  (2.1)

so

−2ν ≤ H_{0} ≤ 2ν  (2.2)

For V, a bounded function on ℤ^{ν}, let

H = H_{0} + V  (2.3)

We are interested in the spectrum of H outside [−2ν, 2ν] = σ(H_{0}).

If we define U on ℓ^{2}(ℤ^{ν}) by

(Uφ)(n) = (−1)^{|n|}φ(n)  (2.4)

where |n| = |n_{1}| + ⋯ + |n_{ν}|, then

UH_{0}U^{−1} = −H_{0}  UV U^{−1} = V  (2.5)

We define, for φ_{+}, φ_{−} ∈ ℓ^{2}(ℤ^{ν}),

Δ(φ_{+}, φ_{−}; V) = ⟨φ_{+}, (H − 2ν)φ_{+}⟩ + ⟨φ_{−}, (−H − 2ν)φ_{−}⟩  (2.6)

Δ > 0 implies that H has spectrum outside [−2ν, 2ν] and, as we will see, Δ(φ_{+}^{(n)}, φ_{−}^{(n)}; V) > 0 for suitable φ_{±}^{(n)} implies the spectral projection χ_{ℝ\[−2ν,2ν]}(H) has infinite dimension.

Note first that

Proposition 2.1. If f, g ∈ ℓ^{2}(ℤ^{ν}), then

Δ(f + g, U(f − g); V) ≥ 2⟨f, (H_{0} − 2ν)f⟩ − 8ν∥g∥^{2} + 4 Re⟨f, V g⟩  (2.7)

Proof. By (2.5),

Δ(f + g, U(f − g); V) = ⟨(f + g), (H_{0} − 2ν + V)(f + g)⟩
+ ⟨(f − g), (H_{0} − 2ν − V)(f − g)⟩
= 2⟨f, (H_{0} − 2ν)f⟩ + 2⟨g, (H_{0} − 2ν)g⟩ + 4 Re⟨f, V g⟩

By (2.2), H_{0} ≥ −2ν, so

⟨g, (H_{0} − 2ν)g⟩ ≥ −4ν∥g∥^{2}

This yields (2.7). □

One obvious choice is to take f = φ, g = γV φ. The V -terms on the right side of (2.7) are then

∥V φ∥^{2}(−8νγ^{2} + 4γ)  (2.8)

which is maximized at γ = \frac{1}{4ν}, where −8νγ^{2} + 4γ = \frac{1}{2ν}. Thus we have a generalization of (1.7).

Theorem 2.2. For any φ ∈ ℓ^{2}(ℤ^{ν}),

Δ((1 + \frac{1}{4ν} V)φ, U(1 − \frac{1}{4ν} V)φ; V ) ≥ 2⟨φ, (H_{0} − 2ν + \frac{1}{4ν} V^{2})φ⟩  (2.9)

In some applications, we will want to be able to estimate ∥f ± g∥ in terms of f, and so want to cut off V g. We have Theorem 2.3. For any F ∈ ℓ^{∞} with 0 ≤ F ≤ 1, we have

Δ(φ(1 + (4ν)^{−1}F V), Uφ(1 − (4ν)^{−1}F V); V ) ≥ 2⟨φ, (H_{0} − 2ν + (4ν)^{−1}F V^{2})φ⟩ (2.10)

Proof. By taking g = γF V φ, f = φ, the V -terms in (2.7) are

−8νγ^{2}∥F V φ∥^{2} + 4γ⟨V φ, F V φ_{−}⟩  (2.11)

in place of (2.8). Since 0 ≤ F ≤ 1, we have −F^{2} ≥ −F, so

−∥F V φ∥^{2} ≥ −⟨V φ, F V φ⟩

and (2.10) results. □

The properties of H_{0} needed above are only (2.2) and (2.5). If J is a Jacobi matrix (1.2) and J_{1} is the Jacobi matrix with the same values of a_{n} but with b_{n} = 0, then UJ_{1}U^{−1} = −J_{1}. (2.2) is replaced by

J_{1} ≥ −α  (2.12)

where

α = max_{n} (a_{n} + a_{n+1})  (2.13)

One has

Theorem 2.4. For any φ ∈ ℓ^{2}(ℤ_{+}), with φ_{±} = (1 ± γV)φ (where γ = (2 + α)^{−1}), we have

⟨φ_{+}, (J − 2)φ_{+}⟩ + ⟨Uφ_{−}, (−2 − J)Uφ_{−}⟩ ≥ 2⟨φ, (J_{1} − 2 + γb^{2})φ⟩  (2.14)

3. A V^{2} Comparison Theorem

Our goal in this section is to prove the following extension of Theorem 5:

Theorem 3.1. Let V be defined on ℤ^{ν}. Let V (n) → 0 as |n| → ∞. If H_{0} + (4ν)^{−1}V^{2} has at least one eigenvalue (resp., infinitely many) outside [−2ν, 2ν], then so does H_{0} + V.

The key to this will be Theorem 2.2, but we will also need

Lemma 3.2. Let W ≥ 0 on ℤ^{ν} with W (n) → 0 as |n| → ∞. If H_{0} + W has infinitely many eigenvalues in (2ν, ∞), then we can find {φ_{n}}_{n=1}^{∞} with ⟨φ_{n}, (H_{0} + W)φ_{n}⟩ > 2ν∥φ_{n}∥^{2}, so that each φ_{n} has finite support and

dist(supp(φ_{n}), supp(φ_{m})) ≥ 2  (3.1)

for all n ≠ m.

Proof. Let Λ_{k} = {n ∈ ℤ^{ν} | max_{i=1,...,ν}|n_{i}| ≤ k}. We first claim that for every k, there exists ψ with ψ = 0 on Λ_{k} so that ⟨ψ, (H_{0} + W)ψ⟩ > 2ν∥ψ∥^{2}. For let H̃_{0} be H_{0} with Dirichlet boundary conditions on ∂Λ_{k}, that is, dropping off-diagonal terms H_{0,ij} with i ∈ Λ_{k}, j ∉ Λ_{k} or vice-versa. H̃_{0} − H_{0} is finite rank, so H̃_{0} + W has infinitely many eigenvalues in (2ν, ∞). But H̃_{0} + W is a direct sum of an operator on ℓ^{2}(Λ_{k}) and one on ℓ^{2}(ℤ^{ν}\Λ_{k}). Since dim ℓ^{2}(Λ_{k}) < ∞, we can find ψ ∈ ℓ^{2}(ℤ^{ν}\Λ_{k}) so ⟨ψ, (H_{0} + W)ψ⟩ = ⟨ψ, (H̃_{0} + W)ψ⟩ > 2ν∥ψ∥^{2}.

Now pick φ_{n} inductively as follows. After picking {φ_{n}}_{n=1}^{N}, we have each φ_{n} has finite support, so there is a Λ_{k} with each φ_{n} = 0 on ℤ^{ν}\Λ_{k}, n = 1, …, N.

By the initial argument, pick ψ_{N+1} vanishing on Λ_{k+1} so that ⟨ψ_{N+1}, (H_{0} + W)ψ_{N+1}⟩ > 2ν⟨ψ_{N+1}, ψ_{N+1}⟩ and ψ_{N+1} = 0 on Λ_{k+1}. Let ψ_{N+1}^{(m)} be finitely supported approximations to ψ_{N+1} which vanish on Λ_{k+1}. By continuity, for some m, ⟨ψ_{N+1}^{(m)}, (H_{0} + W)ψ_{N+1}^{(m)}⟩ > 2ν⟨ψ_{N+1}^{(m)}, ψ_{N+1}^{(m)}⟩. Pick φ_{N+1} = ψ_{N+1}^{(m)}. □

Proof of Theorem 3.1. If H_{0} + (4ν)^{−1}V^{2} has at least one eigenvalue outside [−2ν, 2ν], there exists φ with ⟨φ, (H_{0} + \frac{1}{4ν} V^{2} − 2ν)φ⟩ > 0. By (2.9), H_{0} + V has some eigenvalue outside [−2ν, 2ν].

If H_{0} + (4ν)^{−1}V^{2} has infinitely many eigenvalues, by Lemma 3.2, there exist φ_{n} obeying (3.1) so that ⟨φ_{n}, (H_{0} + \frac{1}{4}V^{2})φ_{n}⟩ > 2ν∥φ_{n}∥^{2}. By (2.9), we can find ψ_{n} with either ⟨ψ_{n}, (H_{0} + V)ψ_{n}⟩ > 2ν∥ψ_{n}∥^{2} or ⟨ψ_{n}, (H_{0} + V)ψ_{n}⟩ < −2ν∥ψ_{n}∥^{2} and supp(ψ_{n}) ⊂ supp(φ_{n}). By (3.1), we have

⟨ψ_{n}, ψ_{m}⟩ = 0 and ⟨ψ_{n}, (H_{0} + V)ψ_{m}⟩ = 0 for n ≠ m

Thus, by the min-max principle, H_{0} + V has an infinity of eigenvalues in either (2ν, ∞) or (−∞, −2ν). □

Using Theorem 2.4 in place of Theorem 2.2, we get

Theorem 3.3. Let J({a_{n}}, {b_{n}}) be the Jacobi matrix (1.2). Suppose a_{n} → 1 and b_{n} → 0 so σ_{ess}(J) = [−2, 2]. Let α be given by (2.13) and γ = (2 + α)^{−1}. If J({a_{n}}, {γb_{n}^{2}}) has at least one eigenvalue (resp., infinitely many) in (2, ∞), then J({a_{n}}, {b_{n}}) has at least one eigenvalue (resp., infinitely many) in (−∞, −2) ∪ (2, ∞).

Remark. In particular, if J({a_{n}}, {b_{n} = 0}) has an infinity of eigenvalues, they cannot be destroyed by a crazy choice of {b_{n}}.

4. Essential Spectra and Compactness in Dimension 1 and 2

Our goal in this section is to prove

Theorem 4.1. Let ν = 1 or 2. If σ_{ess}(H_{0} + V) ⊂ [−2ν, 2ν], then V (n) → 0 as |n| → ∞.

Theorem 4.2. If ν ≥ 3, there exist potentials V in ℓ^{∞}(ℤ^{ν}) so that σ(H_{0} + V) = [−2ν, 2ν] and so that lim sup_{n→∞}|V (n)| > 0.

We will also provide a new proof of Theorem 2.

The key to the dimension dependence is the issue of finding φ_{n} ∈ ℓ^{2}(ℤ^{ν}) so that φ_{n}(0) = 1 and ⟨φ_{n}, (2ν − H_{0})φ_{n}⟩ → 0. We will see that this can be done in dimension 1 and 2. It cannot be done in three or more dimensions, essentially because (2ν −H_{0})^{−1} exists, not as a bounded operator on ℓ^{2} but as a matrix defined on vectors of finite support. To minimize ⟨φ, (2ν−H_{0})φ⟩ subject to φ(0) = 1, by the method of Lagrange multipliers, one takes φ̃ = (2ν − H_{0})^{−1}δ_{0}/⟨δ_{0}(2ν − H_{0})^{−1}δ_{0}⟩. This is not in ℓ^{2} but has ℓ^{2} approximations. In fact, let φ ∈ ℓ^{2} with φ(0) = ⟨δ_{0}, φ⟩ = 1. By the Cauchy-Schwarz inequality, 1 ≤ ∥(2ν −H_{0})^{1/2}φ∥ ∥(2ν −H_{0})^{−1/2}δ_{0}∥, that is,

⟨φ, (2ν − H_{0})φ⟩ ≥ ⟨δ_{0}, (2ν − H_{0})^{−1}δ_{0}⟩^{−1} > 0

for ν ≥ 3. So any ℓ^{2} sequence φ with φ(0) = 1 has a minimal kinetic energy in dimension ν ≥ 3.

A different way of thinking about this is as follows: If φ has compact support in a box of size L and φ(0) = 1, then, on average, ∇φ is at least L^{−1} so ∥∇φ∥^{2} = ⟨φ, (2ν − H_{0})φ⟩ ∼ L^{ν}L^{−2}. If ν ≥ 3, one does not do better by taking big boxes. In ν = 1, one certainly does; and in ν = 2, a careful analysis will give (ln L)^{−1} decay.

Proposition 4.3. Let L_{1}, L_{2} ≥ 1. There exists φ_{L_{1},L_{2}} ∈ ℓ^{2}(ℤ), supported in [−L_{1}, L_{2}], so that

(i) φ_{L_{1},L_{2}}(0) = 1 (ii) ⟨φ_{L_{1},L_{2}}, (2 − H_{0})φ_{L_{1},L_{2}}⟩ = (L_{1} + 1)^{−1} + (L_{2} + 1)^{−1} (iii) for suitable constants c_{1} > 0 and c_{2} < ∞,

c_{1}(L_{1} + L_{2}) ≤ ∥φ_{L_{1},L_{2}}∥^{2} ≤ c_{2}(L_{1} + L_{2})  (4.1)

Proof. Define

φ_{L_{1},L_{2}}(n) = \begin{cases}1 − \frac{n}{L_{2}+1} & 0 ≤ n ≤ L_{2} + 1 \\ 1 − \frac{|n|}{L_{1}+1} & 0 ≤ −n ≤ L_{1} + 1 \\ 0 & n ≥ L_{2} + 1 or n ≤ −L_{1} − 1 \end{cases}  (4.2)

then (i) and (iii) are easy. As

⟨ψ, (2 − H_{0})ψ⟩ = ∑_{j=−∞}^{∞} [ψ(j + 1) − ψ(j)]^{2}  (4.3)

for any ψ ∈ ℓ^{2}(ℤ), we have

⟨φ_{L_{1},L_{2}}, (2 − H_{0})φ_{L_{1},L_{2}}⟩ = ∑_{j=1}^{L_{2}+1}(\frac{1}{L_{2}+1})^{2} + ∑_{j=−1}^{L_{1}+1}(\frac{1}{L_{1}+1})^{2}
= (L_{1} + 1)^{−1} + (L_{2} + 1)^{−1}

which proves (ii). □

Remark. If ψ(0) = 1 and ψ is supported in [−L_{1}, L_{2}],

∑_{j=1}^{L_{2}+1} ψ(j) − ψ(j − 1) = −1

so, by the Schwarz inequality,

1 ≤ (L_{2} + 1) ∑_{j=1}^{L_{2}+1} |ψ(j) − ψ(j − 1)|^{2}

Thus

⟨ψ, (2 − H_{0})ψ⟩ ≥ (L_{1} + 1)^{−1} + (L_{2} + 1)^{−1}

which shows that (4.2) is an extremal function.

Proposition 4.4. Let L ≥ 1. There exists φ_{L} ∈ ℓ^{2}(ℤ^{2}) supported in {(n_{1}, n_{2}) | |n_{1}| + |n_{2}| ≤ L} so that

(i) φ_{L}(0) = 1 (ii) 0 ≤ ⟨φ_{L}, (4 − H_{0})φ_{L}⟩ ≤ c[ln(L + 1)]^{−1} for some c > 0 (iii) (L^{−1} ln(L))^{2}∥φ_{L}∥^{2} → d > 0

Remark. It seems clear that one cannot do better than ln(L)^{−1} in the large L asymptotics of ⟨φ_{L}, (4 − H_{0})φ_{L}⟩ for any test function obeying (i) and the support condition.

Proof. Define

φ_{L}(n_{1}, n_{2}) = \begin{cases}\frac{− ln[(1+|n_{1}|+|n_{2}|)/(L+1)]}{ln(L+1)} & if |n_{1}| + |n_{2}| ≤ L \\ 0 & if |n_{1}| + |n_{2}| ≥ L \end{cases}

then (i) is obvious. As

ln(\frac{a + 1}{(L + 1)}) − ln(\frac{a}{(L + 1)}) = ln(1 + \frac{1}{a}) ≤ a^{−1}

we have that

⟨φ_{L}, (4 − H_{0})φ_{L}⟩ = ∑_{n_{1},n_{2}}(φ_{L}(n_{1} + 1, n_{2}) − φ_{L}(n_{1}, n_{2}))^{2}
+ (φ_{L}(n_{1}, n_{2} + 1) − φ_{L}(n_{1}, n_{2}))^{2}

≤ ln(L + 1)^{−2} ∑_{\begin{matrix}n_{1},n_{2} \\ |n_{1}|+|n_{2}|≤L \end{matrix}} (1 + |n_{1}| + |n_{2}|)^{−2}

≤ c ln(L + 1)^{−1}

since the sum diverges as ln L. This proves (ii).

To prove (iii), we note that, by a simple approximation argument,

ln(L)^{2}L^{−2}∥φ_{L}∥^{2} → ∫∫_{|x|+|y|≤1}[ln(|x| + |y|)]^{2} dx dy

as L → ∞. □

Proof of Theorem 4.1. Consider first the case ν = 1. Suppose lim sup|V (n)| = a > 0. Pick L so that 2(L + 1)^{−1} < \frac{1}{8} min(a^{2}, 2a). Pick a sequence n_{1}, …, n_{j}, … with |V (n_{j})| → a so that |n_{j}| − max_{1≤ℓ≤j−1}|n_{ℓ}| ≥ 2(L + 2). Thus, |n_{j} − n_{ℓ}| ≥ 2(L + 2) for all j ≠ ℓ.

Define

F (n) = min(1, \frac{2}{|V (n)|})  (4.4)

and let ψ_{j}(n) = φ_{L,L}(n − n_{j}). Then

⟨ψ_{j}, (H_{0} − 2 + \frac{1}{4} F V^{2})ψ_{j}⟩ ≥ −2(L + 1)^{−1} + \frac{1}{4} F (n_{j})V (n_{j})^{2}
≥ −\frac{1}{8} min(a^{2}, 2a) + \frac{1}{4} min(|V (n_{j})|^{2}, 2|V (n_{j})|)

Thus we have that

lim inf⟨ψ_{j}, (H_{0} − 2 + \frac{1}{4} F V^{2})ψ_{j}⟩ ≥ \frac{1}{8} min(a^{2}, 2a)

As |F V | ≤ 2, if φ_{±,j} = (1 ± \frac{1}{4} F V)ψ_{j}, we have

\frac{1}{2} ∥ψ_{j}∥ ≤ ∥φ_{±,j}∥ ≤ \frac{3}{2} ∥ψ_{j}∥ ≤ C_{L}  (4.5)

where C_{L} is independent of j; compare (4.1).

By (2.9), we have a subsequence of j’s so that either

lim inf⟨φ_{+,j_{ℓ}}, (H_{0} + V − 2)φ_{+,j_{ℓ}}⟩ ≥ \frac{1}{16} min(a^{2}, 2a)

or

lim inf⟨φ_{−,j_{ℓ}}, (−H_{0} − V − 2)φ_{+,j_{ℓ}}⟩ ≥ \frac{1}{16} min(a^{2}, 2a)

Moreover, the φ’s are orthogonal. Thus H has essential spectrum in either

[2 + \frac{1}{16} d^{−1} min(a^{2}, 2a), ∞) or (−∞, −2 − \frac{1}{16} d^{−1} min(a^{2}, 2a)]

The proof for ν = 2 is similar, using Proposition 4.4 in place of Proposition 4.3. □

Proof of Theorem 4.2. We will give an example with V ≥ 0. Thus the only spectrum that H_{0} + V can have outside [−2ν, 2ν] is in (2ν, ∞).

As ν ≥ 3, the operator (2ν − H_{0})^{−1} has finite matrix elements despite being unbounded. We denote the n, m matrix element, the Green function, by G_{ν}(n−m). By the Birman-Schwinger principle [18, Section 3.5], if the matrix

M_{nm} = V (n)^{1/2}G_{ν}(n − m)V (m)^{1/2}

defines an operator on ℓ^{2}(ℤ^{ν}) with norm strictly less than 1, then H_{0} + V has no spectrum in (2ν, ∞).

Since G_{ν}(n) → 0 as n → ∞ (indeed, it decays as |n|^{−(ν−2)}), we can find a sequence in ℤ^{ν} with |n_{j}| → ∞ and

∑_{j≠k} |G_{ν}(n_{j} − n_{k})| < \frac{1}{2}  (4.6)

For example, pick n_{k} inductively so ∑_{j<k} G_{ν}(n_{j} −n_{k}) < 2^{−k−2}. (Actually, G_{ν}(n− m) > 0 for all n and m so the absolute value sign is redundant.) Choose λ > 0 so that

λG_{ν}(0) < \frac{1}{2}  (4.7)

and define V by

V (n) = \begin{cases}min(1, λ) & n = some n_{j} \\ 0 & otherwise \end{cases}

In this way, lim sup_{|n|→∞}|V (n)| = min(1, λ) > 0. However, by Schur’s lemma, ∥M∥ < 1 so H_{0} + V has no eigenvalues. □

The ideas in the first part of this section allow us to reprove Theorem 2 and, more importantly, extend it to two dimensions.

Theorem 4.5. Let ν = 1 or 2. If σ(H_{0} + V) ⊂ [−2ν, 2ν], then V = 0.

Proof. By Theorems 4.1 and 4.2, V (n) → 0. By Theorem 3.1, if H_{0} + V has no bound states, neither does H_{0}+ \frac{1}{4ν} V^{2}. Since V = 0 if and only if V^{2} = 0, we may as well consider the case V ≥ 0. Let φ_{L} be the function guaranteed by Proposition 4.3 or 4.4. Then

⟨φ_{L}, (H_{0} + V − 2ν)φ_{L}⟩ ≥ V (0) + ⟨φ_{L}, (H_{0} − 2ν)φ_{L}⟩

Since ⟨φ_{L}, (H_{0} − 2ν)φ_{L}⟩ → 0, we must have V (0) = 0. By translation invariance, V (n) = 0 for all n. □

Theorem 4.6. Let J be the Jacobi matrix (1.2). Suppose lim inf a_{n} ≥ 1 and σ_{ess}(J) ⊂ [−2, 2]. Then b_{n} → 0 as n → ∞.

Proof. Since lim inf a_{n} ≥ 1, we can suppose a_{n} ≥ 1 since the change from a_{n} to min(a_{n}, 1) is a compact perturbation. By the lemma below, σ_{ess}(J) can only shrink if a_{n} ≥ 1 is replaced by a_{n} = 1. Thus we can suppose a_{n} = 1 in what follows.

Let H̃ = H_{0} on ℓ^{2}(ℤ\ℤ^{+}) with a Dirichlet boundary condition at 0, H̃ = J on ℓ^{2}(ℤ^{+}), and

V (n) = \begin{cases}0 & n ≤ 0 \\ b_{n} & n ≥ 1 \end{cases}

Then H = H_{0} + V differs from H̃ by a finite rank perturbation. Thus H has essential spectrum in [−2, 2]. The proof is completed by using Theorem 4.1. □

Lemma 4.7. If J({a_{n}}, {b_{n}}) is the Jacobi matrix given by (1.2), then sup σ_{ess}(J({a_{n}}, {b_{n}})) and − inf σ_{ess}(J({a_{n}}, {b_{n}})) are monotone increasing as a_{n} increases.

Proof. As noted in Section 3 of Hundertmark-Simon [6], for each N, the sum of the N largest positive eigenvalues, ∑_{j=1}^{N} E_{j}^{+}(J({a_{n}}, {b_{n}})), is monotone in {a_{n}}. But

sup σ_{ess}(J(\{a_{n}\}, \{b_{n}\})) = lim_{n→∞} \frac{1}{N} ∑_{j=1}^{N} E_{j}^{+}(J(\{a_{n}\}, \{b_{n}\}))

The proof for − inf σ_{ess} is similar. □

5. Decay and Bound States for Half-Line Discrete Schrödinger Operators

While whole-line discrete Schrödinger operators have bound states if V ≢ 0 (Theorem 2), this is not true for half-line operators. Indeed, the discrete analogue of Bargmann’s bound [6] implies that

∑_{n=1}^{∞} n|V (n)| < 1 ⇒ σ(J_{0} + V) = [−2, 2]  (5.1)

where J_{0} is the free Jacobi operator, that is, (1.2) with a_{n} ≡ 1, b_{n} ≡ 0.

One can also include the endpoint case: If a sequence of selfadjoint operators A_{k} converges strongly to A, then

σ(A) ⊆ ⋂_{n} \overline{⋃_{k≥n} σ(A_{k})}

see [13, Theorem VIII.24]. This shows that (5.1) can be extended to

∑_{n=1}^{∞} n|V (n)| ≤ 1 ⇒ σ(J_{0} + V) = [−2, 2]  (5.2)

In this section, we explore what the absence of bound states tells us about the decay of V. We begin with the case V ≥ 0:

Theorem 5.1. Suppose V (n) ≥ 0 and that J_{0} + V has no bound states. Then

|V (n)| ≤ n^{−1}  (5.3)

Moreover, (5.3) cannot be improved in that for each n_{0}, there exists V_{n_{0}} so that V_{n_{0}}(n_{0}) = n_{0}^{−1} and J_{0} + V_{n_{0}} has no bound states.

Proof. Let W_{n_{0}} be

W_{n_{0}}(n) = \begin{cases}1 & n = n_{0} \\ 0 & n ≠ n_{0} \end{cases}

We claim J_{0} + λW_{n_{0}} has a bound state if and only if |λ| > n_{0}^{−1}. By (1.6), we can suppose λ > 0. In that case, by a Sturm oscillation theorem [17], there is a bound state in (2, ∞) if and only if the solution of

u(n + 1) + u(n − 1) + λW_{n_{0}}(n)u(n) = 2u(n)  u(0) = 0, u(1) = 1  (5.4)

has a negative value for some n ∈ ℤ^{+}. The solution of (5.4) is

u(n) = \begin{cases}n & n ≤ n_{0} \\ n_{0} + (1 − λn_{0})(n − n_{0}) & n ≥ n_{0} \end{cases}

which takes negative values if and only if λn_{0} > 1. This proves the claim.

In particular, n_{0}^{−1}W_{n_{0}} = V_{n_{0}} is a potential where equality holds in (5.3) and σ(J_{0} + V_{0}) = [−2, 2].

On the other hand, if V (n_{0}) > n_{0}^{−1}, then since V ≥ 0, V (n) ≥ V (n_{0})W_{n_{0}}(n) for all n and so, by a comparison theorem and the fact that we have shown J_{0} + V (n_{0})W_{n_{0}} has a bound state, we have that J_{0} + V has a bound state. The contrapositive of V (n_{0}) > n_{0}^{−1} ⇒ σ(J_{0} + V) ≠ [−2, 2] is the first assertion of the theorem. □

Remark. Notice that Theorem 5.1 says (5.2) is optimal in the very strong sense that if ∑_{n=1}^{∞} α_{n}|V (n)| ≤ 1 ⇒ σ(J_{0} + V) = [−2, 2] for all potentials V, then each α_{n} ≤ n.

Positivity of the potential made the proof of Theorem 5.1 elementary. Because of the magic of Theorem 5, we can deduce a result for V ’s of arbitrary sign:

Theorem 5.2. If J_{0} + V has no bound states, then

|V (n)| ≤ 2n^{−1/2}  (5.5)

Moreover, (5.5) cannot be improved by more than a factor of 2 in that for each n_{0}, there exists V_{n_{0}} so that J_{0} + V_{n_{0}} has no bound states and

lim_{n_{0}→∞} n_{0}^{1/2}|V_{n_{0}}(n_{0})| = 1

Remarks. (a) The proof shows

V_{n_{0}}(n_{0}) = \sqrt{\frac{1}{n_{0}} + \frac{1}{4n_{0}^{2}}} − \frac{1}{2n_{0}} ≡ β_{n_{0}}

so (5.5) cannot be improved to value better than β_{n_{0}} ∼ n_{0}^{−1/2} − \frac{1}{2}n_{0}^{−1}. (b) In [3] it is shown that the absence of bound states implies

|V (n)| ≤ \sqrt{2}n^{−1/2}(1 + \frac{2}{n})^{3/2}

and that there are examples V_{n_{0}} with V_{n_{0}}(n_{0}) = \sqrt{2}n_{0}^{−1/2} and no bound states.

Proof. Theorem 5 extends to the situation where H_{0} is replaced by J_{0} since the mapping φ → φ(1 ± F V) is local. Thus if J_{0} + V has no bound states, neither does J_{0} + \frac{1}{4}V^{2}. Since V^{2} ≥ 0, Theorem 5.1 applies, and thus \frac{1}{4}|V (n)|^{2} ≤ n^{−1}, which is (5.5).

For the other direction, let W_{n_{0}} be

W_{n_{0}} = \begin{cases}1 & n = n_{0} \\ −1 & n = n_{0} + 1 \\ 0 & n ≠ n_{0}, n_{0} + 1 \end{cases}

A direct solution of (5.4) is

u(n) = \begin{cases}n & n ≤ n_{0} \\ (1 − λ)n_{0} + 1 + (1 + λ − λ^{2}n_{0})(n − n_{0} − 1) & n ≥ n_{0} + 1 \end{cases}  (5.6)

Thus u(n) has a negative value if and only if 1 + λ − λ^{2}n_{0} < 0. Define

λ_{±}^{crit} = ±\sqrt{\frac{1}{4n_{0}^{2}} + \frac{1}{n_{0}}} − \frac{1}{2n_{0}}  (5.7)

If |λ| > min(|λ_{+}^{crit}|, |λ_{−}^{crit}|), u takes negative values for either u(n, λ) or u(n, −λ). By (1.6), J_{0} +V has eigenvalues in (−∞, −2) if and only if J_{0} −V has eigenvalues in (2, ∞). Thus since |λ_{+}^{crit}| < |λ_{−}^{crit}|, J_{0} + λW_{n_{0}} has no eigenvalues if |λ| ≤ λ_{+}^{crit}. □

One can also say something about infinitely many bound states:

Theorem 5.3. (i) If V ≥ 0 and

lim sup_{n→∞} |V (n)|n > 1  (5.8)

then J_{0} + V has infinitely many bound states. (ii) For general V, if lim sup_{n→∞}|V (n)|n^{1/2} > 2, then J_{0} + V has infinitely many

bound states.

Proof. (ii) follows from (i) by Theorem 5. To prove (i), suppose J_{0} + V has only finitely many bound states. Then (J_{0} +V −2)u has only finitely many sign changes, so there is N_{0} with u(n)u(n + 1) > 0 if n > N_{0}. It follows that J_{0} + V with Ṽ(n) = V (n + N_{0}) has no bound states. Thus |Ṽ(n)| ≤ n^{−1}, so lim sup_{n→∞} n|V (n)| ≤ 1. Thus, by contrapositives, (5.8) implies J_{0} +V has infinitely many bound states. □

Example 5.4. Let N be a positive integer and n_{k} = N^{2k}. We consider the sequence u(n) which has slope u(n + 1) − u(n) = N^{−k} for n ∈ [n_{k}, n_{k+1}) and then determine the potential V at the sites n_{k} so that u is the generalized eigenfunction at energy 2. (Constancy of the slope in the intervals (n_{k}, n_{k+1}) implies that the potential vanishes there.) We have

u(n_{k}) = n_{1} + (n_{2} − n_{1})N^{−1} + ⋯ + (n_{k} − n_{k−1})N^{−(k−1)}
= (1 − N^{−1}){N^{2} + N^{3} + ⋯ + N^{k}} + N^{k+1}
= N^{k+1}{1 + N^{−1} − N^{−k}}

and so

V (n_{k}) = \frac{2u(n_{k}) − u(n_{k} + 1) − u(n_{k} − 1)}{u(n_{k})}

= \frac{N^{1−k} − N^{−k}}{N^{k+1}\{1 + N^{−1} − N^{−k}\}}

= \frac{1 − N^{−1}}{1 + N^{−1} − N^{−k}} \frac{1}{n_{k}}

As u is monotone, there are no sign flips. We may conclude that J_{0} + V has no bound states because V (n) ≥ 0. Therefore, taking N → ∞, we see that the 1 in (5.8) is optimal.

A similar argument [19] shows there are examples with lim sup n^{1/2}|V (n)| = 1−ε and no bound states for each ε > 0. Basically, V (n) ≠ 0 for n = n_{k} or n_{k} + 1 and V (n_{k}) = −V (n_{k} + 1) = n_{k}^{−1/2}(1 − ε_{k}) with ε_{k} → ε. Again, n_{k} must grow at least geometrically. □

The examples that saturate Theorems 5.1 and 5.3 are sparse, that is, mainly zero. If V is mainly nonzero and comparable in size, the borderlines change from n^{−1} to n^{−2} for positive V ’s and from n^{−1/2} to n^{−1} for V ’s of arbitrary sign.

Theorem 5.5. Let V ≥ 0. Suppose there exists ε > 0 and n_{k} → ∞ so that (i)

\frac{2}{n_{k}} ∑_{j=n_{k}/2}^{n_{k}} V (j) ≥ εV (n_{k})  (5.9)

(ii) lim sup_{k→∞} εn_{k}^{2}V (n_{k}) > 48 Then J_{0} + V has infinitely many bound states.

Proof. For notational simplicity, we suppose each n_{k} is a multiple of 4. By passing to a subsequence, we can suppose that

\frac{εn_{k}}{8} V (n_{k}) > \frac{6}{n_{k}}  (5.10)
\frac{n_{k+1}}{4} > \frac{3}{2} n_{k} + 2  (5.11)

Let u_{k} be the function which is 1 at n_{k}, has constant slope on the intervals [\frac{n_{k}}{4} − 1, n_{k}] and [n_{k}, \frac{3n_{k}}{2} + 1], and vanishes at n = \frac{n_{k}}{4} − 1 and n = \frac{3n_{k}}{2} + 1. By Proposition 4.3,

⟨u_{k}, (2 − J_{0})u_{k}⟩ ≤ \frac{6}{n_{k}}

On [\frac{n_{k}}{2} , n_{k}], we have |u(j)| ≥ \frac{1}{2}, so

⟨u_{k}, V u_{k}⟩ ≥ \frac{1}{4} ∑_{j=n/2}^{n} V (j) ≥ \frac{εn_{k}}{8} V (n_{k})  (by (5.9))

By (5.10), ⟨u_{k}, (J_{0} + V − 2)u_{k}⟩ > 0 for all k. By (5.11) for k ≠ ℓ,

⟨u_{k}, u_{ℓ}⟩ = ⟨u_{k}, (J_{0} + V)u_{ℓ}⟩ = 0

so, by the min-max principle, J_{0} + V has infinitely many eigenvalues in (2, ∞). □

Theorem 5 and Theorem 5.5 immediately imply

Theorem 5.6. Suppose there exists ε > 0 and n_{k} → ∞ so that

(i) \frac{2}{n_{k}} ∑_{j=n_{k}/2}^{n_{k}}|V (j)|^{2} ≥ ε^{2}|V (n_{k})|^{2} (ii) lim sup_{k→∞} εn_{k}|V (n_{k})| > 8\sqrt{3} Then J_{0} + V has infinitely many bound states.

In this regard, here is another application of Theorem 5:

Theorem 5.7. If |V (n)| ≥ \frac{β}{n} with β > 1 and V (n) → 0, then J_{0} + V has infinitely many bound states.

Proof. It is known (see [2, Theorem A.7]) if β^{2} > 1, then the operator with potential \frac{β^{2}}{4n^{2}} , and hence the operator with potential \frac{1}{4}V (n)^{2} ≥ \frac{β^{2}}{4n^{2}} , has infinitely many bound states. The assertion now follows from Theorem 5. □

Corollary 5.8. If V (n) → 0 but lim inf_{|n|→∞} |nV (n)| > 1, then J_{0} + V has infinitely many bound states. The same result holds in the whole-line setting.

Proof. We begin with the half-line case. By hypothesis, there exists a β > 1 such that |V (n)| ≥ \frac{β}{n} for all but finitely many n. Therefore the claim follows from the previous theorem because a finite rank perturbation can remove at most finitely many eigenvalues. The whole-line case follows by Dirichlet decoupling. □

Remark. It is known (see [2]) that if V (n) = \frac{1}{4n^{2}} or V (n) = β \frac{(−1)^{n}}{n} with |β| < \frac{1}{2}, then J_{0} + V has finitely many bound states. Thus the powers n^{−2} and n^{−1} in the previous results are optimal.

The optimal constant in Theorem 5.7 is 1, as we now show.

Proposition 5.9. For β ∈ [−1, 1], the operator J_{0} + V with potential V (n) = β \frac{(−1)^{n}}{n} has no bound states.

Proof. We will show that the operator with potential V (n) = \frac{(−1)^{n}}{n} has no bound states. As the absolute value of a bound state eigenvalue is an increasing function of the coupling constant, this implies that potentials of the form V (n) = β \frac{(−1)^{n}}{n} have no bound states for β ∈ [0, 1]. Equation (2.5) shows that J_{0} + V is unitarily equivalent to −(J_{0} − V). Thus, the proposition for β ∈ [−1, 0] follows from the β ∈ [0, 1] case.

By the unitary equivalence of J_{0} + V and −(J_{0} − V), it suffices to show that for V_{0} = (−1)^{n}/n, J_{0} + V_{0} and J_{0} − V_{0} have no eigenvalues in (2, ∞).

We look at solutions of

u(n + 1) + u(n − 1) = (2 ∓ V_{0}(n))u(n)  (5.12)

By Sturm oscillation theory, the number of eigenvalues of J_{0} ± V_{0} in (2, ∞) is equal to the number of zeros, in (0, ∞), of the linear interpolation of the generalized eigenfunction—that is, the solution of (5.12) with u(0) = 0. Moreover, the Sturm separation theorem implies that if (5.12) has a solution with u(n) > 0 for n = 0, 1, 2, ..., then the generalized eigenfunction must be positive for n ≥ 1 (and not ℓ^{2}; see remark below).

We are able to write down positive solutions explicitly, but rather than pull such a rabbit out of a hat, we provide some explanation. Motivated by calculations in Maple, we look for solutions with u(n) = u(n + 1) for either all odd n or all even n. This is equivalent to asking if

( \begin{matrix}x & −1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}y & −1 \\ 1 & 0 \end{matrix} ) = ( \begin{matrix}xy − 1 & −x \\ y & −1 \end{matrix} )  (5.13)

has \binom{1}{1} as an eigenvector. If this is true for y = E−V (n), x = E−V (n+1) for all odd (resp. even) n, then the Schrödinger equation has a solution with u(n) = u(n − 1) for all odd (resp. even) n, and for such n,

u(n + 2) = [E − V (n) − 1]u(n)  (5.14)

The matrix in (5.13) has \binom{1}{1} as an eigenvector if and only if

xy = x + y  (5.15)

If x = 2 + a, y = 2 + b, then (5.12) becomes

ab = −a − b  (5.16)

This is solved by b = \frac{1}{m}, a = −\frac{1}{m+1} with y − 1 = 1 + \frac{1}{m}. Since −V (n) appears in the transfer matrix for V_{0}, we take m = 2n + 1, n = 0, 1, 2, … and find a solution with

u(0) = u(1) = 1 u(2n) = u(2n + 1), u(2n + 2) = (1 + \frac{1}{2n+1})u(2n)

which is a positive solution with u(n) → ∞ as n^{1/2} as n → ∞. For −V_{0}, we take m = 2n, n = 1, 2, …, and find a solution with

u(0) = 0 u(1) = u(2) = 1 u(2n) = u(2n − 1) u(2n + 2) = (1 + \frac{1}{2n})u(2n)

so again, u(n) → ∞ as n^{1/2}. We have thus found the required solution to show J_{0} + V_{0} has no eigenvalues in (2, ∞). □

Remarks. (a) It follows from the proof that the generalized eigenfunctions at energies ±2 are not square summable. This shows that ±2 are not eigenvalues.

(b) Choosing y = −\frac{1}{m}, x = \frac{1}{m+1} in the arguments given above shows that there are solutions u_{±} of (J_{0} + V_{0})u = 0 with |u_{±}(n)| ∼ |n|^{±1/2} as n → ∞. This shows that 0 is not an eigenvalue of J_{0} + V_{0} but suggesting that for J_{0} + (1 + ε)V_{0}, there are solutions ℓ^{2} at infinity for ε > 0. That is, just as coupling 1 is the borderline for eigenvalues outside [−2, 2], it is the borderline for an eigenvalue at E = 0 similar to the Wigner-von Neumann phenomenon.

As our final topic, we want to discuss divergence of eigenvalue moments if |V (n)| ∼ n^{−α} with α < 1.

Lemma 5.10. Let A be a bounded selfadjoint operator. Let {φ_{j}}_{j=1}^{∞} be an orthonormal set with

⟨φ_{j}, Aφ_{k}⟩ = α_{j}δ_{jk}  (5.17)

If F is a nonnegative even function on ℝ that is monotone nondecreasing on [0, ∞), then

Tr(F (A)) ≥ ∑_{j} F (α_{j})  (5.18)

Remarks. (a) As F (A) ≥ 0, it follows that Tr(F (A)) is always defined although it may be infinite.

(b) In particular, if φ_{j} is a family of nonzero vectors in ℓ^{2}(ℤ^{+}) with dist(supp(φ_{j}), supp(φ_{k})) ≥ 2 for j ≠ k, then for J = J_{0} + V,

Tr(F (J)) ≥ ∑_{j} F (|\frac{⟨φ_{j}, Jφ_{j}⟩}{⟨φ_{j}, φ_{j}⟩}|)  (5.19)

Proof. Let E_{1} ≥ E_{2} ≥ ⋯ be the eigenvalues of |A|. By min-max and max-min for A, we have E_{j} ≥ |α_{j}|^{∗} where |α_{j}|^{∗} is the decreasing rearrangement of |α_{j}|. So (5.18) follows. □

Lemma 5.11. Let |V | ≤ 4ν on supp(φ). Then there exists ψ with supp(ψ) = supp(φ) so that

∥ψ∥^{−2}|⟨ψ, (H_{0} + V)ψ⟩| − 2ν ≥ \frac{1}{4} [∥φ∥^{−2}⟨φ, (H_{0} + \frac{1}{4ν} V^{2})φ⟩ − 2ν]  (5.20)

Proof. Let ψ_{±} = (1 ± (4ν)^{−1}V)φ. Since |V | ≤ 4ν, ∥ψ_{±}∥^{2} ≤ 4∥φ∥^{2}. The result now follows from (2.9) by choosing ψ to be either ψ_{+} or Uψ_{−}. □

Theorem 5.12. Let J be a Jacobi matrix of the form J_{0} + V where

|V (n)| ≥ Cn^{−α}  (5.21)

for some α < 1 and V (n) → 0. Then

∑_{j} (|E_{j}| − 2)^{γ} = ∞  (5.22)

for

γ < \frac{1 − α}{2α}  (5.23)

where E_{j} are eigenvalues of J outside [−2, 2].

Remark. In particular, the eigenvalue sum ∑_{j=1}^{∞}(|E_{j}| − 2)^{1/2} critical of Szegőtype sum rules [7, 16] diverges if α < \frac{1}{2}. This illuminates results in [2, 16].

Proof. Fix p > 0. Let φ_{m} be supported near m^{p+1} on an interval [m^{p+1} − C_{1}m^{p}, m^{p+1} + C_{1}m^{p}] where C_{1} is picked to arrange that supports are separated by at least 2. Taking the slopes fixed on each half-interval and using Proposition 4.3, we see

⟨φ_{m}, (2 − H_{0})φ_{m}⟩ ≤ \frac{C_{2}}{m^{p}}  (5.24)
⟨φ_{m}, \frac{1}{4} V^{2}φ_{m}⟩ ≥ \frac{C_{3}m^{p}}{m^{2α(p+1)}}  (5.25)
⟨φ_{m}, φ_{m}⟩ ≥ C_{4}m^{p}  (5.26)

So long as α(p + 1) < p (i.e., p < \frac{α}{1−α}), (5.25) beats out (5.24) for large m, and we find

⟨φ_{m}, φ_{m}⟩^{−1}⟨φ_{m}, (H_{0} + \frac{1}{4} V^{2} − 2)φ_{m}⟩ ≥ C_{5}m^{−2α(p+1)}  (5.27)

As p ↓ \frac{α}{1−α}, 2α(p + 1) ↓ \frac{2α}{1−α}.

By the lemma with F (x) = dist(x, [−2, 2])^{γ}, we see that we have divergence if (5.23) holds. □

Remarks. (a) If the constant C in (5.21) is large enough, we can take p = \frac{α}{1−α} and get divergence if γ = \frac{1−α}{2α}.

(b) One can extend this result as well as Theorems 5.3 and 5.5 to higher dimensions.

References

[1] M. Cwikel, Weak type estimates for singular values and the number of bound states of Schrödinger operators, Ann. of Math. 106 (1977), 93–100.

[2] D. Damanik, D. Hundertmark, and B. Simon, Bound states and the Szegő condition for Jacobi matrices and Schrödinger operators, preprint.

[3] D. Damanik and R. Killip, in preparation.

[4] P. Deift and B. Simon, Almost periodic Schrödinger operators, III. The absolutely continuous spectrum in one dimension, Comm. Math. Phys. 90 (1983), 389–411.

[5] S. Denisov, On Nevai’s conjecture and Rakhmanov’s theorem for Jacobi matrices, preprint [6] D. Hundertmark and B. Simon, Lieb-Thirring inequalities for Jacobi matrices, J. Approx.

Theory 118 (2002), 106–130.

[7] R. Killip and B. Simon, Sum rules for Jacobi matrices and their applications to spectral theory, to appear in Ann. of Math.

[8] M. Klaus, On the bound state of Schrödinger operators in one dimension, Ann. Phys. 108 (1977), 288–300.

[9] L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Non-relativistic Theory. Course of Theoretical Physics, Vol. 3, Addison-Wesley, Reading, Mass., 1958.

[10] E.H. Lieb, Bounds on the eigenvalues of the Laplace and Schrödinger operators, Bull. Amer. Math. Soc. 82 (1976), 751–753; see also The number of bound states of one-body Schrödinger operators and the Weyl problem, in “Geometry of the Laplace Operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), pp. 241–252, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980.

[11] P. Nevai, Weakly convergent sequences of functions and orthogonal polynomials, J. Approx. Theory 65 (1991), 322–340.

[12] E.A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials, II, Math. USSR Sb. 46 (1983), 105–117.

[13] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, New York, 1980.

[14] G.V. Rozenblum, Distribution of the discrete spectrum of singular differential operators, Dokl. Akad. Nauk SSSR 202 (1972), 1012–1015); Izv. VUZaved. Matematika 1 (1976), 75–86. [Russian]

[15] B. Simon, The bound state of weakly coupled Schrödinger operators in one and two dimensions, Ann. Phys. 97 (1976), 279–288.

[16] B. Simon and A. Zlatoš, Sum rules and the Szegő condition for orthogonal polynomials on the real line, preprint

[17] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathematical Surveys and Monographs Vol. 72, American Mathematical Society, Providence, R.I., 2000.

[18] W. Thirring, A Course in Mathematical Physics. Vol. 3. Quantum Mechanics of Atoms and Molecules, Lecture Notes in Physics, 141. Springer-Verlag, New York-Vienna, 1981.

[19] A. Zlatoš, private communication.