BOUND STATES AND THE SZEGŐ CONDITION FOR JACOBI MATRICES AND SCHRÖDINGER OPERATORS

DAVID DAMANIK^{1,2}, DIRK HUNDERTMARK^{1}, AND BARRY SIMON^{1,3}

Abstract. For Jacobi matrices with a_{n} = 1 + (−1)^{n}αn^{−γ}, b_{n} = (−1)^{n}βn^{−γ}, we study bound states and the Szegő condition. We provide a new proof of Nevai’s result that if γ > \frac{1}{2}, the Szegő condition holds, which works also if one replaces (−1)^{n} by cos(μn). We show that if α = 0, β ≠ 0, and γ < \frac{1}{2}, the Szegő condition fails. We also show that if γ = 1, α and β are small enough (β^{2} + 8α^{2} < \frac{1}{24} will do), then the Jacobi matrix has finitely many bound states (for α = 0, β large, it has infinitely many).

1. Introduction

This paper focuses on Jacobi matrices, that is, operators J on ℓ^{2}(ℤ_{+}), where ℤ_{+} = {1, 2, … }, given by (b_{n} real, a_{n} > 0)

(Ju)(n) = a_{n}u(n + 1) + b_{n}u(n) + a_{n−1}u(n − 1)  (1.1)

where the a_{n−1}u(n − 1) term is dropped if n = 1. We define J_{0} by a_{n} ≡ 1, b_{n} ≡ 0, and will suppose J −J_{0} is compact, so σ_{ess}(J) = [−2, 2]. We are interested especially in the Szegő condition,

Z(J) ≡ \frac{1}{2π} ∫_{−2}^{2} log(\frac{\sqrt{4 − E^{2}}}{2π \frac{dν_{ac}}{dE}} )\frac{dE}{\sqrt{4 − E^{2}}} < ∞  (1.2)

where ν is the spectral measure for J and the vector δ_{1}. We will also consider some aspects of Schrödinger operators −Δ + V.

In 1979, Nevai [24] proved a conjecture of Askey that if

a_{n} = 1 + \frac{(−1)^{n}α}{n} + O(n^{−2}) b_{n} = \frac{(−1)^{n}β}{n} + O(n^{−2})  (1.3)

then the Szegő condition holds. Our goal here is to understand this result from the point of view of sum rules recently used to study the Szegő condition by Killip-Simon [17] and Simon-Zlatoš [31], and to consider various extensions and borderline cases, in particular, the following four questions:

Date: August 19, 2002.

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

E-mail: damanik@its.caltech.edu; dirkh@its.caltech.edu; bsimon@caltech.edu.

2 Supported in part by NSF grant DMS-0227289.

3 Supported in part by NSF grants DMS-9707661, DMS-0140592.

(1) Nevai [24] allows replacement of \frac{(−1)^{n}}{n} by \frac{(−1)^{n}}{n^{γ}} with γ > \frac{1}{2} and still gets (1.2). Is γ = \frac{1}{2} a borderline or just where Nevai’s method fails? We will see that γ = \frac{1}{2} is indeed a borderline and that a_{n} = 1, b_{n} = \frac{(−1)^{n}}{n^{γ}} obeys (1.2) if and only if γ > \frac{1}{2}. This is a subtle issue: one might think the key is that b_{n+1} − b_{n} decay faster than n^{−1}, in which case γ = \frac{1}{2} is not special but, as is the case in many other situations [18], b_{n} ∈ ℓ_{2} is critical; see Theorem 2 below.

(2) What is the condition on the errors O(n^{−2}) in (1.3)? Nevai actually shows if those errors, e_{a}(n), e_{b}(n), obey ∑_{n=2}^{∞}(log n)|e_{j}(n)| < ∞ for j ∈ {a, b}, then (1.2) still holds. In line with the advances in [17, 31], we will only require ∑_{n=2}^{∞}|e_{j}(n)| < ∞, for j ∈ {a, b}. Indeed, our results are logarithmically better than Nevai’s in the leading term. If \frac{(−1)^{n}}{n} in (1.3) is replaced by \frac{(−1)^{n}}{n^{1/2}} [log n]^{−γ}, then Nevai’s method requires γ > 1, while we require only γ > \frac{1}{2}.

(3) What about other oscillatory potentials like \frac{cos(ηn)}{n^{γ}} for η ∈ (0, 2π)? (1.3) is the case η = π. Although it is possible his methods extend to this case, the conditions in Nevai’s paper require cancellations in b_{n} + b_{n+1} and do not work for η ≠ π. We will accommodate general η.

(4) Nevai’s work suggests that \frac{(−1)^{n}}{n} is akin to n^{−2} potentials, which suggests that for |α|+|β| small, (1.3) has finitely many eigenvalues outside [−2, 2] while for |α| + |β| large, it has infinitely many. We will prove the finiteness result below. We note that while he does not discuss this case explicitly, Chihara’s conditions in [7] imply finitely many bound states if a_{n} = 1 and |β| is small.

For Jacobi matrices, our main results are:

Theorem 1. Suppose

a_{n} = 1 + c_{n} + d_{n+1} − d_{n}  (1.4)
b_{n} = e_{n} + f_{n+1} − f_{n}  (1.5)

with

∑_{n=1}^{∞} |c_{n}| + |e_{n}| + |d_{n}|^{2} + |f_{n}|^{2} < ∞  (1.6)

Then Z(J) < ∞ and

∑_{j,±} \sqrt{E_{j}^{±}(J)^{2} − 4} < ∞  (1.7)

where E_{j}^{±}(J) are the eigenvalues of J in ±(2, ∞).

Remarks. 1. In case a_{n} − 1 = \frac{α(−1)^{n}}{n^{γ}} + e_{a}(n), b_{n} = \frac{β(−1)^{n}}{n^{γ}} + e_{b}(n), we define

d_{n} = − ∑_{j=n}^{∞} \frac{α(−1)^{n}}{n^{γ}}  f_{n} = − ∑_{j=n}^{∞} \frac{β(−1)^{n}}{n^{γ}}  (1.8)

Since the sums are O(n^{−γ}), (1.6) is then true if ∑|e_{a}(n)| + |e_{b}(n)| < ∞ and γ > \frac{1}{2}.

2. If b_{n} is instead \frac{cos(ηn)}{n}, it is still true that f_{n} ≡ − ∑_{j=n}^{∞} \frac{cos(ηn)}{n} is O(n^{−1}), and so in ℓ^{2}, and thus this theorem also includes cases like \frac{cos(ηn)}{n} where b_{n} + b_{n+1} does not have cancellations.

3. By mimicking the construction of Wigner and von Neumann (see, e.g., [27, Example 1 in Chapter XIII.13], one can construct Jacobi matrices J with a_{n} ∼ 1 + \frac{(−1)^{n}}{n} and b_{n} ∼ \frac{(−1)^{n}}{n} as n → ∞ which have 0 as an eigenvalue embedded in the essential spectrum.

As a converse to Theorem 1, we note

Theorem 2. Suppose

(i) lim sup[− ∑_{j=1}^{n} log(a_{j})] > −∞ (ii) ∑_{n=1}^{∞}(a_{n} − 1)^{2} + b_{n}^{2} = ∞ Then Z(J) = ∞.

Remark. If a_{n} = 1 (or a_{n} = exp(\frac{α(−1)^{n}}{n^{γ}})), b_{n} = \frac{β(−1)^{n}}{n^{γ}}, and γ ≤ \frac{1}{2}, then this implies Z(J) = ∞, showing γ = \frac{1}{2} is the borderline.

Proof. Suppose Z(J) < ∞ and (i) holds. Then, by Theorem 1 of Simon-Zlatoš [31], (1.7) holds. A fortiori, the quasi-Szegő condition, (1.8) of Killip-Simon [17], and the \frac{3}{2} Lieb-Thirring bound hold. So, by Theorem 1 of Killip-Simon [17], (ii) fails. Thus, (i) + Z(J) < ∞ ⇒ not (ii). So (i) + (ii) ⇒ Z(J) = ∞. □

Theorem 3. Suppose a_{n} = 1 and (1.5) holds with

lim sup_{n} n^{2}[|e_{n}| + |f_{n}|^{2} + |f_{n+1}|^{2}] < \frac{1}{8}  (1.9)

Then J has only finitely many bound states. If (1.4), (1.5) hold and

lim sup_{n} n^{2}[|c_{n}| + |c_{n−1}| + 24|d_{n−1}|^{2} + 48|d_{n}|^{2} + 24|d_{n+1}|^{2}
+ |e_{n}| + 6|f_{n}|^{2} + 6|f_{n+1}|^{2}] < \frac{1}{8}  (1.10)

then J has finitely many bound states.

Remark. In particular, if a_{n} = 1, b_{n} = \frac{β(−1)^{n}}{n}, and |β| < \frac{1}{2}, then J has only finitely many bound states. If |β| > 1, it is proven in [11] that J has infinitely many bound states. Also, if a_{n} = 1 + \frac{α(−1)^{n}}{n} and b_{n} as before, then for β^{2} + 8α^{2} < \frac{1}{24}, J has also only finitely many bound states, but this bound seems to be far from optimal.

The techniques we will use are two-fold: First, we will use the result of Simon-Zlatoš [31] that if − ∑_{j=1}^{∞} log(a_{j}) is conditionally convergent, then (1.2) holds if and only if (1.7) holds (by a Case-type sum rule). This means that all the results on finiteness on Z(J) which we are discussing are equivalent to suitable bounds on eigenvalues. Second, to bound eigenvalues, we will use ideas developed in the 1970’s to discuss Schrödinger operators with oscillatory potentials [1, 5, 6, 8, 9, 15, 16, 23, 29, 30, 32]. Interestingly, the focus of that work was to handle wild, pathological cases like V (r) = (1 + r)^{−2}e^{1/r} sin(e^{1/r}) or V (r) = (1 + r)^{−2}e^{r} sin(e^{r}), which are extremely unbounded near r = 0 or r = ∞ but whose oscillators cause −Δ + V, defined by quadratic form methods, to be well behaved. In fact, we believe that the most interesting examples are ones like r^{−1} sin(r) which are not unbounded at all, but oscillatory and slowly decaying.

Most of the 1970’s papers discuss scattering or selfadjointness results, although Combescure-Ginibre [9] and Chadan-Martin [6] do discuss bounds on the number of bound states. Since they were not as efficient in using operator bounds, we begin in Section 2 with the continuum Schrödinger operator case. In Section 3, we discuss the growth of N(λV) as λ → ∞ for long-range oscillatory potentials. We will prove

Theorem 4. Let V_{β}(x) = (1 + |x|)^{−β} sin(|x|) for 2 > β > 1. On ℝ^{ν}, we have

−a_{−} + b_{−}λ^{ν/β} ≤ N(λV) ≤ a_{+} + b_{+}λ^{ν/β}

for suitable (β-dependent) a_{±}, b_{±} > 0.

We note that if β > 2 so V_{β} ∈ L^{ν/2}, then it is known (see [27, Theorem XIII.80]) that

lim_{λ→∞} λ^{−ν/2}N(λV) = \frac{τ_{ν}}{(2π)^{ν}} ∫_{V_{β}(x)≤0}(−V_{β}(x))^{ν/2} d^{ν}x

with τ_{ν} the volume of the unit ball in ℝ^{ν}.

In Section 4, we discuss the discrete Schrödinger case, that is, Jacobi matrices with a_{n} ≡ 1. In Section 5, we discuss the general Jacobi case. The appendix contains bounds on the O(n^{−2}) situation that we will need in the body of the paper. Since these have not been proven in the Jacobi case with optimal constants, it was necessary to include this appendix. In particular, in Theorems A.6 and A.7, we study Jacobi matrices J with |a_{n} − 1| ∼ \frac{γ_{a}}{n^{2}} and |b_{n}| ∼ \frac{γ_{b}}{n^{2}} and discuss finiteness (resp. infinitude) of the discrete spectrum of J in [−2, 2]^{c}, depending on whether 2γ_{a} +γ_{b} < \frac{1}{4}, (resp. 2γ_{a} +γ_{b} > \frac{1}{4}), thereby extending results of Chihara in [7].

We would like to thank Rowan Killip, Paul Nevai, Mihai Stoiciu, and Andrej Zlatoš for valuable communications.

2. The Continuum Schrödinger Case

Let W be an ℝ^{ν}-valued C^{1} function on ℝ^{ν} or a piecewise C^{1} continuous function on ℝ so that ∇·W is also bounded. In fact, once one has the bounds below, it is easy to accommodate arbitrary distributions W with W ∈ L^{ν} + L^{∞} (when ν ≥ 3) even if ∇ · W is not bounded. For our applications of interest, we make these simplifying assumptions.

Proposition 2.1 (Combescure-Ginibre Lemma [9]). If φ ∈ C_{0}^{∞},

|⟨φ, ∇W φ⟩| ≤ 2∥W φ∥ ∥∇φ∥  (2.1)

Proof. First, integrate by parts, ⟨φ, ∇W φ⟩ = 2 Re⟨W φ, ∇φ⟩. Then use the Schwarz inequality. □

Theorem 2.2. Let H = −Δ + V_{1} + ∇ · W and H_{1} = −Δ + 2V_{1} − 4W^{2}. Then

H ≥ \frac{1}{2} H_{1}  (2.2)

In particular, if N(V) is the number of negative eigenvalues of −Δ+V, then

N(V_{1} + ∇ · W) ≤ N(2V_{1} − 4W^{2})  (2.3)

and if E_{p}(V) = Tr([−HE_{(−∞,0]}(H)]^{p}) for H = −Δ + V is the Lieb-Thirring sum of eigenvalue powers, then

E_{p}(V_{1} + ∇ · W) ≤ 2^{−p}E_{p}(2V_{1} − 4W^{2})  (2.4)

Proof. By (2.1),

⟨φ, (−Δ + V_{1} + ∇W)φ⟩ ≥ ⟨φ, (−Δ + V_{1})φ⟩ − ε⟨φ, −Δφ⟩ − ε^{−1}⟨φ, W^{2}φ⟩

= ⟨φ, (1 − ε)(−Δ + \frac{1}{1 − ε} V_{1} − \frac{1}{ε(1 − ε)} W^{2})φ⟩

In the absence of a V_{1} term, the optimal choice of ε is ε = \frac{1}{2} (to minimize \frac{1}{ε(1−ε)}), so we make that choice in general. It yields (2.2), which in turn immediately implies (2.3) and (2.4). □

The bound (2.1) and its proof are taken from Combescure-Ginibre [9]. While they use the Schwarz inequality, they do not explicitly note (2.2), which causes them to make extra arguments that can be less efficient than using (2.2). For example, if V_{1} = 0, ν = 3, and

w^{2} = (4π)^{−2} ∫ d^{3}x dy^{3} \frac{W^{2}(x)W^{2}(y)}{|x − y|^{2}}

then (2.3) and the Birman-Schwinger principle immediately imply that

N(∇ · W) ≤ 16w^{2}

while Combescure-Ginibre [9] only claim

N(∇ · W) ≤ 16w^{2}(1 + w)^{2}

which is much worse for large w.

For ν = 1, V_{1} = 0, (2.3) is a result of Chadan-Martin [6] who use Sturm comparison methods rather than the Schwarz inequality and the Combescure-Ginibre lemma. Theorem 2.2 has some immediate consequences:

Corollary 2.3. For ν ≥ 3,

N(V_{1} + ∇ · W) ≤ c_{ν}(∥V_{1}∥_{ν/2}^{ν/2} + ∥W ∥_{ν}^{ν})  (2.5)

For general ν and p ≥ \frac{1}{2} if ν = 1, p > 0 if ν = 2, and p ≥ 0 if ν ≥ 3,

E_{p}(V_{1} + ∇W) ≤ c_{ν,p}(∥V_{1}∥_{p+ν/2}^{p+ν/2} + ∥W ∥_{2p+ν}^{2p+ν})  (2.6)

Proof. (2.5) is just the Cwikel-Lieb-Rozenblum [10, 19, 28] bound, given (2.3). (2.6) is the Lieb-Thirring bound [21, 22] when p is strictly larger than the minimal value. p = 0 for ν ≥ 3 is (2.5) while p = \frac{1}{2}, ν = 1 is due to Weidl [34] (see also Hundertmark-Lieb-Thomas [13]). □

If ν ≥ 3 and V ∈ L^{ν/2}, we have N(λV) ≤ cλ^{ν/2}, but (2.5) only implies that

N(λ(V_{1} + ∇W)) ≤ c_{1}λ^{ν/2} + c_{2}λ^{ν}

In the next section, we see that in some specific cases, N(λV) really does grow at rates arbitrarily close to λ^{ν}.

Corollary 2.4. If ν ≠ 2 and 4W^{2} − 2V_{1} ≤ \frac{(ν−2)^{2}}{4} |x|^{−2}, then −Δ + V_{1} + ∇W_{1} has no bound states. If 4W^{2} < \frac{(ν−2)^{2}}{4} |x|^{−2} and V_{1} ∈ L^{ν/2} (if ν ≥ 3) or ∫ (1 + |x|)|V_{1}(x)| dx < ∞ (ν = 1), then −Δ + V_{1} + ∇ · W_{1} has finitely many bound states.

Proof. The first statement is immediate from (2.3) and Theorem A.3. The second follows from

−Δ + 2V_{1} − 4W^{2} = −(1 − ε)Δ − 4W^{2} − εΔ + 2V_{1}

and the Cwikel-Lieb-Rozenblum estimate if ν ≥ 3 and Bargmann’s bound if ν = 1. □

Example 2.5. If V = \frac{sin(r)}{r^{α}}, we can write V = V_{1} + ∇ · W where W = r̂(−\frac{cos(r)}{r^{α}} − \frac{α sin(r)}{r^{α+1}})f(r) where f ∈ C^{∞} vanishes near 0 and is 1 near ∞. This shows that for α < 1, −Δ+λV_{0} has finitely many bound states for all λ, and when α = 1, it has finitely many bound states if |λ| is small. An argument similar to that in [11] shows that if α = 1 and λ is large, −Δ + λV has an infinity of negative eigenvalues. □

3. Schrödinger Operators at Large Coupling

Our purpose here is to prove Theorem 4 that V_{β} = \frac{sin(r)}{(1+r)^{β}} (with 2 > β > 1) has N(λV_{β}) growing as λ^{ν/β} for λ large. We give the details when ν = 1 on a half-line and then discuss the case when ν ≥ 2.

Proof of Theorem 4. Half-line case: We begin with the upper bound. Let φ_{R}(r) be a C^{∞} function with ∥φ_{R}∥_{∞} = 1, which is 0 if r < R and 1 if r > R + 1. By translation, we may assume the derivatives \frac{d^{α}φ_{R}}{dx^{α}} are uniformly bounded in x and R (for fixed α). Let W_{R}(r) = − ∫_{r}^{∞} φ_{R}(s)V_{β}(s) ds and V_{1,R} = V_{β} − \frac{d}{dr}W_{R}. Define V_{2,R}(r) = max_{s≥r} |V_{1,R}(s)|. Then

−\frac{d^{2}}{dx^{2}} + λV_{β} = (−\frac{1}{2} \frac{d^{2}}{dx^{2}} + λV_{1,R}) + (−\frac{1}{2} \frac{d^{2}}{dx^{2}} + λ \frac{dW_{R}}{dr} )
≥ (−\frac{1}{2} \frac{d^{2}}{dx^{2}} − λV_{2,R}) + \frac{1}{2} (−\frac{1}{2} \frac{d^{2}}{dx^{2}} − 8λ^{2}W_{R}^{2}) (3.1)

Next, note that

W_{R}(r) ≤ C(max(r, R))^{−β}  (3.2)

We also have

|V_{2,R}(r)| ≤ r^{−β}  (3.3)

and is zero if r > R + 1.

Calogero [3] has proven that if V is monotone decreasing and nonnegative, then N(−V) ≤ 2π^{−1} ∫_{0}^{∞}|V (s)|^{1/2} ds. This bound, (3.1)– (3.3), and the fact that dim(E_{(−∞,0)}(A + B)) ≤ dim E_{(−∞,0)}(A) + dim E_{(−∞,0)}(B) (by the variational principle) imply that for any R,

N(λV_{β}) ≤ C_{1}[λ^{1/2} ∫_{0}^{R+1} r^{−β/2} dr + λ ∫_{0}^{∞} max(R, r)^{−β} dr]
= C_{2}[λ^{1/2}R^{1−β/2} + λR^{1−β}]

since 1 < β < 2. Pick R = λ^{1/β} and get

N(λV_{β}) ≤ 2C_{2}λ^{1/β}

On the other side, consider the operator H̃(λ), which is −\frac{d^{2}}{dx^{2}} + λV_{β} with Dirichlet boundary conditions added at the points (2n + \frac{3}{2})π ± \frac{π}{3}. Adding such boundary conditions only increases the operator, so N(λV_{β}) ≥ # of negative eigenvalues of H̃(λ). In each interval of the form [(\frac{2n+3}{2})π − \frac{π}{3}, (\frac{2n+3}{2})π + \frac{π}{3}], sin(r) is less than −\frac{1}{2}, so V_{β} ≤ −\frac{λ}{2[(2n+3)π]^{β}} on the entire interval. The lowest Dirichlet eigenvalue of −\frac{d^{2}}{dx^{2}} on such an interval is \frac{9}{4}, so each interval with

\frac{λ}{2[(2n + 3)π]^{β}} > \frac{9}{4}

contributes an eigenvalue so

N(λV_{β}) ≥ C_{3}λ^{1/β}

This completes the proof of Theorem 4 in the half-line case.

One might think that it would help to use the fact that small n intervals provide O(λ^{1/2}) eigenvalues rather than just the 1 we use, but a detailed analysis shows it improves the constant in front of λ^{1/β} but not the power.

Higher dimensions: The lower bound is similar to the half-line case. We have sin(r) < −\frac{1}{2} on annuli which we can partially cover with suitable disjoint cubes of fixed size, finding cubes where V is deep enough when the distance of the cube from the origin is no more than Cλ^{1/β}. The number of such cubes is O(λ^{ν/β}) so we get an O(λ^{ν/β}) lower bound.

For the upper bound when ν ≥ 3, we can replace Calogero’s bound with the Cwikel-Lieb-Rozenblum bound. Since ∫_{0}^{R} r^{−βν/2} d^{ν}r = C_{4}R^{ν(1−β/2)} and ∫_{R}^{∞} r^{−βν} d^{ν}r = C_{5}R^{ν(1−β)}, we find

N(λV_{β}) ≤ C_{6}[λ^{ν/2}R^{ν(1−β/2)} + λ^{ν}R^{ν(1−β)}]

so picking R = λ^{1/β}, we get N(λV_{β}) ≤ C_{7}λ^{ν/β}.

ν = 2 is messier. We will sketch the idea, but omit the details. One needs to use the spherical symmetry and consider each partial wave separately. By using the analog of (3.1), we see, on functions of angular momentum ℓ, there is an effective potential which bounds −Δ + λV_{β} from below, viz,

V_{ℓ,eff} = −\frac{1}{4r^{2}} + \frac{ℓ^{2}}{4r^{2}} − λr^{−β}χ_{(0,R+1)} − λ^{2} max(r, R)^{−2β}

We need to consider three regions:

(i) ℓ ≥ C_{8}λ^{1/β}: Take R = λ^{1/β} and find V_{ℓ,eff} ≥ 0 so there are no bound states.

(ii) 1 ≤ ℓ ≤ C_{8}λ^{1/β}: We take R = λ^{1/β}, drop the \frac{ℓ^{2}−4}{4r^{2}} term, and use Calogero’s bound to get a bound per partial wave of C_{9}λ^{1/β} as in the one-dimensional case.

(iii) ℓ = 0: The singularity of −r^{−2} at both 0 and infinity requires us to place Dirichlet boundary conditions at 1 and a point R_{2} = λ^{2/β−1}, which for large λ is much larger than R_{1} = λ^{1/β} (since \frac{1}{β} < 1 < \frac{1}{β−1} < \frac{2}{β−1}). On (R_{2}, ∞), we can use the fact that λV_{β} ≥ −\frac{1}{4} \frac{1}{r^{2} log r} and Theorem A.2 to see the Dirichlet operator has no bound states. On (0, 1), we can bound the ℓ = 0 states by all states for the Dirichlet Laplacian in L^{2}({|x| < 1}, d^{2}x) with Dirichlet boundary conditions with energy below c_{a}λ (where c = max_{|r|≤1} −\frac{sin(r)}{(1+r)^{β}}). It is known (by Weyl’s theorem, see [27, p. 271]) that this is asymptotically c_{10}λ since ν = 2. In (1, R_{2}), we can use Calogero’s bound where now “V ” is −\frac{1}{4r^{2}} −λV_{2,R}−4λ^{2}W_{R}^{2}. We get a bound by c_{11} ∫_{1}^{R_{2}} \frac{dr}{r} + c_{12}λ^{1/β}. Taking into account the possible two states lost by adding the Dirichlet boundary conditions in the ℓ = 0 space, we get

N(λV_{β}) ≤ c_{13}λ^{1/β}(λ^{1/β}) + c_{14}(λ + λ^{1/β} + log(|λ| + 1))

which is the required large λ^{2/β} bound.

□

4. Discrete Schrödinger Operators

Our main goal in this section is to extend (2.1) and Theorem 2.2 to the discrete case. It will be convenient to consider operators on all of ℤ and get bounds on Jacobi matrices by restriction. We will also restrict to eigenvalues above energy 2. One can then control energies below −2 by using

U_{0}J({a_{n}}, {b_{n}})U_{0}^{−1} = −J({a_{n}}, {−b_{n}})  (4.1)

where J({a_{n}}, {b_{n}}) is the Jacobi matrix (1.1) with parameters a_{n}, b_{n}, and

(U_{0}u)(n) = (−1)^{n}u(n)  (4.2)

On ℓ^{2}(ℤ), define two operators H_{0} and δ_{+} as

(H_{0}u)(n) = u(n + 1) + u(n − 1)  (4.3)
(δ_{+}u)(n) = u(n + 1) − u(n)  (4.4)

Then δ_{−} ≡ δ_{+}^{∗} is given by

(δ_{−}u)(n) = u(n − 1) − u(n)

and

δ_{+}δ_{−} = δ_{−}δ_{+} = 2 − H_{0}  (4.5)

(for if δ_{+} = R−1 and δ_{−} = L−1, then RL = LR = 1 and H_{0} = L+R). Let b_{n} and f_{n} be sequences on ℤ and suppose

b_{n} = f_{n+1} − f_{n} = (δ_{+}f)_{n}  (4.6)

Then in ℓ^{2}(ℤ), for u real and of finite support,

⟨u, bu⟩ = ∑_{n} b_{n}|u(n)|^{2}

= ∑_{n}(f_{n+1} − f_{n})|u(n)|^{2}

= ∑_{n} f_{n}(|u(n − 1)|^{2} − |u(n)|^{2})

= ⟨δ_{−}u, f(1 + L)u⟩  (4.7)

Since ∥δ_{−}u∥^{2} = ⟨u, δ_{+}δ_{−}u⟩ = ⟨u, (2 − H_{0})u⟩ by (4.5), we see that

Lemma 4.1. If b is given by (4.6), then

|⟨u, bu⟩| ≤ ⟨u, (2 − H_{0})u⟩^{1/2}[2⟨u, (f^{2} + f̃^{2})u⟩]^{1/2}  (4.8)

where

f̃_{n} = f_{n+1}  (4.9)

Proof. In getting (4.8), we used (4.5), (4.7),

∥f(1 + L)u∥^{2} ≤ 2∥fu∥^{2} + 2∥fLu∥^{2} = 2∥fu∥^{2} + 2∥f̃u∥^{2}

and the fact that, because of ||x| − |y|| ≤ |x − y|, we also have

⟨|u|, (2 − H_{0})|u|⟩ = \frac{1}{2} ∑_{n}||u(n + 1)| − |u(n)||^{2} ≤ ⟨u, (2 − H_{0})u⟩

so it suffices to prove the result for real valued sequences u. □

We will later need the following estimate that was proven along the way (we get J_{0} by restricting to u’s of support on ℤ_{+}):

|⟨u, δ_{+}f u⟩| ≤ 2|⟨u, (2 − H_{0})u⟩|^{1/2}|⟨u, \frac{1}{2} (f^{2} + f̃^{2})u⟩|^{1/2}  (4.10)
≤ ε⟨u, (2 − J_{0})u⟩ + ε^{−1}⟨u, \frac{1}{2} (f^{2} + f̃^{2})u⟩  (4.11)

Theorem 4.2. Let b_{n} be a sequence on ℤ_{+} so that lim_{n→∞} ∑_{j=1}^{n} b_{j} exists, and let

f_{n} = − ∑_{j=n}^{∞} b_{j}  (4.12)

Let J be the Jacobi matrix with a_{n} ≡ 1 and b’s given by b_{n}. Let J^{±} be the Jacobi matrix with a_{n} ≡ 1 and b’s given by

±2(f^{2} + f̃^{2})  (4.13)

Then (i)

dim E_{(2,∞)}(J) ≤ dim E_{(2,∞)}(J^{+})  (4.14)

(ii)

dim E_{(−∞,−2)}(J) ≤ dim E_{(−∞,−2)}(J^{−})  (4.15)

(iii) If the eigenvalues E of J^{±} outside [−2, 2] obey

∑_{j}(|E_{j}(J^{±})| − 2)^{α} < ∞  (4.16)

for some α and for both J^{+} and J^{−}, then

∑_{j}(|E_{j}(J)| − 2)^{α} < ∞  (4.17)

Proof. Define

b_{0} = − ∑_{j=1}^{∞} b_{j}

so if f is extended to ℤ by setting f_{k} = 0 for k ≤ 0, we have b = f̃− f. Thus, by (4.8) as operator on ℓ^{2}(ℤ),

−b ≥ \frac{1}{2} [−(2 − H_{0}) − 2(f^{2} + f̃^{2})]

so

2 − H_{0} − b ≥ \frac{1}{2} [(2 − H_{0}) − 2(f^{2} + f̃^{2})]

Now restrict to functions supported on ℤ_{+} to get

2 − J ≥ \frac{1}{2} [2 − J^{+}]  (4.18)

This yields (4.14) and (4.1) then yields (4.15). The two together imply (4.17). □

Example. Let b_{n} = \frac{β(−1)^{n}}{n}. Then f_{n} ∼ −\frac{1}{2} \frac{β(−1)^{n}}{n} + O(\frac{1}{n^{2}} ) and the leading term in 2n^{2}(f^{2} + f̃^{2}) = 4(\frac{1}{2}β)^{2} = β^{2}. By Theorem A.6, if β^{2} < \frac{1}{4}, J(a = 1, b) has finitely many eigenvalues, that is, |β| < \frac{1}{2} produces finitely many eigenvalues.

On the other hand, if |β| > 1, it is known [11] that H has an infinite number of bound states. It would be interesting to determine the exact value of the coupling constant, where the shift from finitely many to infinitely many bound states takes place.

Proof of the First Assertion in Theorem 3. By (4.18), if b_{n} has the form (1.4) and J^{±} is formed with b_{n}^{±} = 2e_{n}^{±} ± 2(f_{n}^{2} + f_{n+1}^{2}), then

2 ∓ J ≥ \frac{1}{2} [2 ∓ J^{±}]

If (1.9) holds, then

lim sup_{n} n^{2}[|b̃_{n}^{±}|] < \frac{1}{4}

so J^{±} have finitely many bound states by Theorem A.6. □

5. Oscillatory Jacobi Matrices

In this section, we will prove Theorems 1, 2, and 3 by accommodating general values of a_{n} within the bounds of the last section. Recall that R acts as Ru(n) = u(n + 1) and we defined δ_{+} = R − 1. It will be convenient to write the Jacobi matrix J = aR + R^{∗}a + b in “divergence form,” that is, write it as J = −δ_{+}^{∗}gδ_{+} + q. Let us first consider the whole-line case: Given sequences a, b on ℤ_{+}, we extend them to sequences on ℤ by setting a_{n} = 1 and b_{n} = 0 for n ≤ 0. We denote the corresponding operator on ℓ^{2}(ℤ) by K. With a^{♯} = R^{∗}aR, that is, a_{n}^{♯} = a_{n−1}, and −δ_{+}^{∗}gδ_{+} = −R^{∗}gR + gR + R^{∗}g − g, we see that g = a and q = b + a + a^{♯}. Thus, recalling δ_{+}^{∗}δ_{+} = 2 − H_{0},

K = −δ_{+}^{∗}aδ_{+} + b + a + a^{♯}
= H_{0} + b + a + a^{♯} − 2 − δ_{+}^{∗}(a − 1)δ_{+}

which shows

⟨u, Ku⟩ = ⟨u, H_{0}u⟩ + ⟨u, (b + a + a^{♯} − 2)u⟩ − ⟨δ_{+}u, (a − 1)δ_{+}u⟩

By restriction to u’s supported on ℤ_{+}, we get

⟨u, Ju⟩ = ⟨u, J_{0}u⟩ + ⟨u, (b + a + a^{♯} − 2)u⟩ − ⟨δ_{+}u, (a − 1)δ_{+}u⟩ (5.1)

where one should keep in mind that a_{1}^{♯} = 1.

We first estimate the third term in (5.1). Writing a as in (1.4), that is, a = 1 + c + δ_{+}d, it reads

⟨δ_{+}u, (a − 1)δ_{+}u⟩ = ⟨δ_{+}u, cδ_{+}u⟩ + ⟨δ_{+}u, (δ_{+}d) δ_{+}u⟩  (5.2)

With (x)_{−} = max(−x, 0), the negative part, we have

⟨δ_{+}u, cδ_{+}u⟩ ≥ −⟨δ_{+}u, c_{−}δ_{+}u⟩

= − ∑_{n}(c_{n})_{−}(|u(n + 1)|^{2} + |u(n)|^{2} − 2 Re(\overline{u(n + 1)}u(n)))

≥ − ∑_{n} 2(c_{n})_{−}(|u(n + 1)|^{2} + |u(n)|^{2})

= −2⟨u, (c_{−} + c_{−}^{♯})u⟩  (5.3)

where one should keep in mind that c_{1}^{♯} = 0. For the last term in (5.2), we note that by (4.11),

⟨δ_{+}u, (δ_{+}d)δ_{+}u⟩ ≥ −⟨δ_{+}u, Aδ_{+}u⟩

where

A = A_{ε} = ε(2 − J_{0}) + \frac{1}{2ε} (d^{2} + d̃^{2})

Now since A ≥ 0,

⟨δ^{+}u, Aδ^{+}u⟩ = ∥A^{1/2}(R − 1)u∥^{2}
≤ 2∥A^{1/2}Ru∥^{2} + 2∥A^{1/2}u∥^{2}
= ⟨u, [2(R^{∗}AR) + 2A]u⟩

We have R^{∗}J_{0}R = J_{0}, R^{∗}f̃R = f, and R^{∗}fR = f^{♯}. Thus we arrive at

⟨δ_{+}u, (δ_{+}d)δ_{+}u⟩ ≥ −⟨u, Bu⟩  (5.4)

with

B = 4ε(2 − J_{0}) + ε^{−1}[(d^{♯})^{2} + 2d^{2} + d̃^{2}]

Writing b = e + δ_{+}f, as in (1.5), and putting (5.1), (5.2), (5.3), and (5.4) together, we have

⟨u, (2 − J)u⟩ ≥(1 − 4ε)⟨u, (2 − J_{0})u⟩
− ⟨u, (e + |c| + |c^{♯}| + ε^{−1}((d^{♯})^{2} + 2d^{2} + d̃^{2}))u⟩
− ⟨u, (δ_{+}f + (δ_{+}d)^{♯} + δ_{+}d)u⟩  (5.5)

Estimating the last term in (5.5) again with the help of (4.11) yields

⟨u, (2 − J)u⟩ ≥(1 − (μ + ν + 4ε))⟨u, (2 − J_{0})u⟩

− ⟨u, (e + |c| + |c^{♯}|)u⟩ − \frac{1}{2ν} ⟨u, (f^{2} + f̃^{2})u⟩

− (\frac{1}{μ} + \frac{1}{ε})⟨u, ((d^{♯})^{2} + 2d^{2} + d̃^{2})u⟩  (5.6)

Choosing μ = ν = ε = \frac{1}{12}, we get

2 − J ≥ \frac{1}{2} [2 − J_{0} − W]  (5.7)

where

W = 2e + 2|c| + 2|c^{♯}| + 12[(f)^{2} + (f̃)^{2}] + 48[(d^{♯})^{2} + 2(d)^{2} + (d̃)^{2}] (5.8)

Proof of Theorem 1. (1.6) and (5.8) imply

∑ |W_{n}| < ∞

Thus, by Hundertmark-Simon [14], (1.7) holds for the eigenvalues of J_{0} ± W_{0}. By (5.7), (4.1), and the min-max principle, (1.7) holds for J.

Moreover, by (1.3) and (1.6), ∑(a_{n} − 1) is conditionally convergent and, by (1.6), ∑(a_{n} − 1)^{2} < ∞. It follows that ∑ log(a_{n}) is conditionally convergent. Thus, by Theorem 1 of Simon-Zlatoš [31], Z(J) < ∞. □

Proof of Theorem 3. By (5.8) and (1.10), for large n,

|W_{n}| ≤ \frac{1 − ε}{4n^{2}}

for some ε > 0. It follows, by Theorem A.6, that J_{0} ± W has only finitely many bound states. Hence, by (4.1), (5.7), and the min-max principle, J has finitely many bound states. □

Appendix A. Finiteness of the Eigenvalue Spectrum for Potentials of a Definite Sign

We need information on finiteness results for nonoscillatory potentials. For Schrödinger operators, these results are well known, but we include some discussion here for two reasons: Optimal constants for Jacobi matrices are not known. The weak L^{ν/2} results we discuss are new. We begin with a version of Hardy’s inequality with optimal constant:

Theorem A.1. Let H_{0} = −\frac{d^{2}}{dx^{2}} on L^{2}(0, ∞) with u(0) = 0 boundary conditions. Let V be a bounded function on [0, ∞) with V (x) → 0 at infinity. Then

(i) If V (x) ≥ (4x^{2})^{−1} for all (resp. all large) x, then H_{0} + V has no (resp. finitely many) bound states. In particular, for any φ ∈ Q(H_{0}), the form domain of H_{0},

∫ \frac{|φ(x)|^{2}}{4x^{2}} dx ≤ ∫ |∇φ(x)|^{2} dx  (A.1)

This is known as Hardy’s inequality. (ii) If V (x) ≤ −(1 + ε)(4x^{2})^{−1} for x > R_{0} and some R_{0}, ε > 0, then

H_{0} + V has an infinite number of bound states.

Remark. We only assume V bounded to avoid technicalities. In fact, one can use (A.1) to discuss V ’s with V (x) ≥ −\frac{1}{4}x^{−2} − c.

Proof. Sturm’s theory (see [27, pp. 90–94]) says that the number of negative eigenvalues of H_{0} + V is precisely the number of zeros of −u′′(x) + V (x)u(x) = 0, u(0) = 0, and that any other solution of −w′′ + V w = 0 has a zero between any two zeros of u, and vice-versa. Thus, if some solution, w, of −w′′ + V w = 0, is positive, H_{0} + V has no eigenvalues, and if it has an infinity of zeros, H_{0} + V has an infinity of eigenvalues.

u(x) = x^{1/2} solves −u′′ − \frac{1}{4}x^{−2}u = 0, showing (i). On the other hand, u(x) = x^{α} with α(α − 1) = \frac{1}{4}(1 + ε) solves −u′′ − \frac{1}{4}(1 + ε)x^{−2}u = 0. If ε > 0, α has an imaginary part and Re(x^{α}) has an infinity of zeros. This plus a comparison theorem implies the results. □

Remark. There are two other ways to prove Hardy’s inequality: Let a = \frac{d}{dx} − \frac{1}{2x}. Then a^{∗}a = H_{0} − (4x^{2})^{−1}; a careful version of this proof requires consideration of boundary conditions at x = 0. Second (Herbst [12]), (A.1) is equivalent to x^{−1}p^{−2}x^{−1} ≤ 4. Changing variables from x to e^{u} = x, using the explicit form p^{−2}(x, y) = max(x, y) for the kernel of p^{−2} with Dirichlet boundary conditions at zero, one gets that x^{−1}p^{−2}x^{−1} is unitarily equivalent to convolution with e^{− \frac{1}{2} |u|} on L^{2}(ℝ). This operator has norm ∫_{−∞}^{∞} e^{− \frac{1}{2} |u|} du = 4. This argument also shows that the operator has continuous spectrum, so if its norm is larger than 1, a Birman-Schwinger-type argument provides an alternate proof of an infinity of bound states.

For reasons that will become clear when we discuss the twodimensional case, we need more on the borderline −\frac{1}{4}x^{−2} case.

Theorem A.2. Let

X_{γ}(x) = −\frac{1}{4x^{2}} − γχ_{(2,∞)}(x) \frac{1}{x^{2}(log x)^{2}}  (A.2)

Let V be a bounded function on [0, ∞) with V (x) → 0 as x → ∞. Then (i) If V (x) ≥ X_{γ=1/4}(x) for large x, then H_{0} + V has finitely many

bound states. (ii) If V (x) ≤ X_{γ}(x) for some γ > \frac{1}{4} and all large x, then H_{0} + V has

infinitely many bound states.

Proof. Let u_{α} = x^{1/2}(log x)^{α} in the region x > 2. Then

−u′′ + X_{γ=−α(α−1)}u = 0

by a direct calculation. The proof is now identical to that of Theorem A.1. □

Theorem A.3. Let H_{0} = −Δ on L^{2}(ℝ^{ν}). Let V be a bounded function on ℝ^{ν} with V (x) → 0 as |x| → ∞. Then

(i) If ν ≥ 3 or on L^{2}([0, ∞)) with Dirichlet boundary conditions at 0 and V (x) ≥ −\frac{(ν−2)^{2}}{4} |x|^{−2}, then −Δ + V has no negative spectrum. If this holds for all |x| > R_{0}, then −Δ + V has finite negative spectrum. In any event, one has Hardy’s inequality,

\frac{(ν − 2)^{2}}{4} ∫ \frac{|φ(x)|^{2}}{|x|^{2}} d^{ν}x ≤ ∫ |∇φ(x)|^{2} d^{ν}x  (A.3)

(ii) If ν = 2 and V (x) ≥ −\frac{1}{4}(|x| log|x|)^{−2} for all |x| ≥ R_{0}, then −Δ+V

has finite negative spectrum. (iii) If ν ≠ 2 and V (x) ≤ −(1+ε)\frac{(ν−2)^{2}}{4} |x|^{−2} for |x| ≥ R_{0}, then −Δ+V

has infinite negative spectrum. (iv) If ν = 2 and V (x) ≤ −(1 + ε)\frac{1}{4}(|x| log|x|)^{−2} for all |x| > R_{0}, then

−Δ + V has an infinite negative spectrum.

Proof. By the min-max principle, it suffices to consider the case where V is spherically symmetric. In that case, −Δ+V is unitarily equivalent (see [26, pp. 160–161]) to a discrete sum ⊕H_{ℓ,m} on ⊕L^{2}([0, ∞), dr) where

H_{ℓ,m} = −\frac{d^{2}}{dr^{2}} + \frac{(ν − 1)(ν − 3)}{4} \frac{1}{r^{2}} + \frac{κ_{ℓ}}{r^{2}} + V

where κ_{ℓ=0} = 0 and all other κ’s have κ_{ℓ} > 0 and κ_{ℓ} → ∞. Since \frac{(ν−1)(ν−3)}{4} − \frac{1}{4} = \frac{(ν−2)^{2}}{4} , this result follows from the previous two theorems. □

The following result seems to be new:

Theorem A.4. Let ν ≥ 3. Let V (x) be a function on ℝ^{ν} so that for any α, m(α) ≡ |{x | |V (x)| > α}| is finite. Suppose

lim_{α↓0} α^{ν/2}m(α) < τ_{ν}(\frac{ν − 2}{2})^{ν}  (A.4)

where τ_{ν} is the volume of the unit ball in ℝ^{ν}. Then −Δ+V has a finite number of bound states.

Proof. Let V_{0} = \frac{(ν−2)^{2}}{4} \frac{1}{|x|^{2}} . m_{0}(α) ≡ |\{x | V_{0}(x) > α\}| = τ_{ν}(\frac{ν−2}{2})^{ν}α^{−ν/2} so (A.4) implies V = V_{1} + V_{2} where V_{1} ∈ L^{ν/2} and V_{2} has a spherical rearrangement, V_{2}^{∗} (see [20]) with

V_{2}^{∗} ≤ (1 − ε)V_{0}^{∗}

for some ε > 0. Now −Δ+V = ε(−Δ)+V_{1} +(1−ε)[−Δ+(1−ε)^{−1}V_{2}]. By the Cwikel-Lieb-Rozenblum [10, 19, 28] bound, ε(−Δ)+V_{1} has finite negative spectrum. By (A.3),

∥V_{0}^{1/2}(−Δ)^{−1}V_{0}^{1/2}∥ ≤ 1

The Brascamp-Lieb-Luttinger inequality [2] shows

∥|V_{2}|^{1/2}(−Δ)^{−1}|V_{2}|^{1/2}∥ ≤ ∥V_{2}^{∗1/2}(−Δ)V_{2}^{∗1/2}∥

It follows that −Δ + (1 − ε)^{−1}V_{2} has no negative spectrum. □

We note that the main part of this paper has results that extend some of these results to Schrödinger operators with oscillatory potentials; see Theorem 2.2. We now turn to the discrete Jacobi case, beginning with

Theorem A.5. Let J_{0} be the free Jacobi matrix. Then

∑_{n=1}^{∞} \frac{1}{4n^{2}} |u(n)|^{2} ≤ (u, (2 − J_{0})u)  (A.5)

Remarks. 1. We will see below that \frac{1}{4} is the optimal constant in this inequality, that is, it is false if \frac{1}{4} is replaced by a larger constant.

2. However, \frac{1}{4n^{2}} can be replaced by \frac{1}{4n^{2}} + \frac{5}{32n^{4}} or, more generally, [(1 + \frac{1}{n})^{1/2} + (1 − \frac{1}{n})^{1/2} − 2].

Proof. There is a Sturm theory in the discrete case [25, 33]. One needs to look at zeros of the linear interpolation of u. In particular, if b_{n} is such that there is a positive solution u_{0} of

(J_{0} + b)u_{0} = 2u_{0}  (A.6)

then (2 − J_{0} − b) ≥ 0. Let u_{0}(n) = n^{1/2} for n ≥ 0. Define for n ≥ 1,

b_{n} = \frac{u_{0}(n + 1) + u_{0}(n − 1)}{u_{0}(n)} − 2 = (1 + \frac{1}{n})^{1/2} + (1 − \frac{1}{n})^{1/2} − 2

Thus (A.6) is obeyed so (2−J_{0} −b) ≥ 0 or ∑ b_{n}|u(n)|^{2} ≤ (u, (2−J_{0})u) for any u. Since (1 − x)^{1/2} = 1 − ∑_{n=1}^{∞} c_{n}x^{n} with c_{n} ≥ 0 and c_{2} = \frac{1}{8}, b_{n} ≥ \frac{1}{4n^{2}} . □

Theorem A.6. Let J be a Jacobi matrix with

lim sup n^{2}|a_{n} − 1| = γ_{a}  (A.7)
lim sup n^{2}|b_{n}| = γ_{b}  (A.8)

both finite with

2γ_{a} + γ_{b} < \frac{1}{4}  (A.9)

Then J has finitely many bound states outside [−2, 2].

Remarks. 1. As we will see, the \frac{1}{4} in (A.9) cannot be improved.

2. In [7], Chihara proves J has finitely many eigenvalues if

lim sup(n^{2}[(a_{n}^{2} − 1) ± \frac{1}{2} (b_{n} + b_{n−1})]) < \frac{1}{16}  (A.10)

(We take his Jacobi matrix and multiply by 2 to get the [−2, 2] rather than [−1, 1] normalization; then his c_{n} and λ_{n} are related to ours by a_{n} = \sqrt{4λ_{n+1}}, b_{n} = 2c_{n}.) This leads to 2γ_{a} + γ_{b} < \frac{1}{16}, so our result, which is best possible, is better by a factor of 4.

3. Because having no eigenvalues remains true if the a_{n}’s are decreased, (A.7) can be replaced by n^{2}(a_{n} − 1)_{+}, although it still must be true that a_{n} → 1 as n → ∞.

Proof. By (4.1), it suffices to prove the spectrum above 2 is finite. Pick ε so that

\frac{2γ_{a} + γ_{b} + 3ε}{1 − ε} ≤ \frac{1}{4}  (A.11)

By changing a_{n} and b_{n} on a finite set (which, because it is a finite rank perturbation of J, cannot change the finiteness of the number of eigenvalues), we can assume for all n,

|a_{n−1} − 1| + |a_{n} − 1| ≤ \frac{2(γ_{a} + ε)}{n^{2}} a_{n} − 1 ≥ −ε |b_{n}| ≤ \frac{γ_{b} + ε}{n^{2}}

(A.12)

By (5.1), we then have

(u, (2 − J)u) ≥ (1 − ε)(u, (2 − J_{0})u) − ∑_{n=1}^{∞} \frac{2γ_{a} + γ_{b} + 3ε}{n^{2}} u_{n}^{2}

≥ (1 − ε)[(u, (2 − J_{0})u) − \frac{1}{4} ∑_{n=1}^{∞} \frac{u_{n}^{2}}{n^{2}} ] ≥ 0

where we first use (A.11) and then (A.5). □

In the other direction, we have

Theorem A.7. Let J be a Jacobi matrix with

lim inf n^{2}(a_{n} − 1) = γ_{a}  (A.13)
lim inf n^{2}b_{n} = γ_{b}  (A.14)

with γ_{a} ≥ 0, γ_{b} ≥ 0, and

2γ_{a} + γ_{b} > \frac{1}{4}  (A.15)

Suppose also that

lim_{n→∞} |a_{n} − 1| + |b_{n}| = 0

Then J has an infinity of eigenvalues in [2, ∞).

Remark. The existence of O(\frac{1}{n^{2}} ) potentials with an infinity of eigenvalues evoked some interest because Case [4] claimed that if sup n^{2}[|a_{n} − 1| + |b_{n}|] < ∞, there were only finitely many eigenvalues. Chihara [7] produced a counterexample with b_{n} ∼ \frac{1}{2n^{2}} , a_{n} − 1 ∼ \frac{3}{8n^{2}} (after changing to our normalization), so 2γ_{a} + γ_{b} = \frac{5}{4}, larger than the needed \frac{1}{4} our theorem allows.

Proof. If there are only finitely many eigenvalues, the solution of (J − 2)u = 0 with u(0) = 0 has only finitely many zeros so, by restricting to the region beyond the zeros and using Sturm theory, we see there is an N_{0} so

u(n) = 0, n ≤ N_{0} ⇒ ⟨u, (2 − J)u⟩ ≥ 0  (A.16)

Define ã_{n} = min(a_{n}, 1 + \frac{γ_{a}}{n^{2}} ), b̃_{n} = min(b_{n}, \frac{γ_{b}}{n^{2}} ) so

lim[n^{2}(ã_{n} − 1) + n^{2}(ã_{n−1} − 1) + n^{2}b̃_{n}] = 2γ_{a} + γ_{b} > \frac{1}{4}  (A.17)

By (5.1) and (A.16) if u(n) = 0 for n ≤ N_{0},

0 ≤ (u, (2 − J)u)

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

≤ ∑_{n=1}^{∞} a_{n}(u(n) − u(n + 1))^{2} + ∑_{n=1}^{∞}(−b̃_{n} − (ã_{n} − 1) − (ã_{n−1} − 1)]u(n)^{2} (A.18)

since b_{n} ≥ b̃_{n}, etc.

Given φ ∈ C_{0}^{∞}(0, ∞) and ℓ = 1, 2, …, pick

u_{n}^{ℓ} = \sqrt{ℓ} φ(\frac{n}{ℓ})  (A.19)

Since supp(φ) is a compact subset of (0, ∞), u_{n}^{(ℓ)} = 0 if n ≤ εℓ for some ε > 0, so (A.18) holds for ℓ large. Since a_{n} → 1,

∑_{n=1}^{∞} a_{n}(u_{n}^{(ℓ)} − u_{n+1}^{(ℓ)})^{2} = ∑_{n=1}^{∞} a_{n}(\frac{φ(\frac{n}{ℓ}) − φ(\frac{n+1}{ℓ})}{\frac{1}{ℓ}})^{2} \frac{1}{ℓ}

→ ∫ φ′(x)^{2} dx

Similarly, by (A.17),

∑_{n=1}^{∞}[−b̃_{n}−(ã_{n} − 1) − (ã_{n−1} − 1)]u_{n}^{2}

= ∑_{n=1}^{∞} n^{2}[−b̃_{n} − (ã_{n} − 1) − (ã_{n−1} − 1)] \frac{φ(\frac{n}{ℓ})^{2}}{(\frac{n}{ℓ})^{2}} \frac{1}{ℓ}

→ − ∫ (\frac{2γ_{a} + γ_{b}}{x^{2}} )φ(x)^{2} dx

we thus have that

−\frac{d^{2}}{dx^{2}} − \frac{2γ_{a} + γ_{b}}{x^{2}} ≥ 0

violating Theorem A.1(ii). This contradiction proves that J must have infinitely many eigenvalues. □

References

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22 D. DAMANIK, D. HUNDERTMARK, AND B. SIMON

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