LIEB-THIRRING INEQUALITIES FOR JACOBI MATRICES

DIRK HUNDERTMARK^{1} AND BARRY SIMON^{1,2}

Abstract. For a Jacobi matrix J on ℓ^{2}(ℤ_{+}) with Ju(n) = a_{n−1}u(n −
1) + b_{n}u(n) + a_{n}u(n + 1), we prove that

∑_{|E|>2}(E^{2} − 4)^{1/2} ≤ ∑_{n}|b_{n}| + 4 ∑_{n}|a_{n} − 1|.

We also prove bounds on higher moments and some related results in higher dimension.

1. Introduction

Let J be a Jacobi matrix, that is, a tridiagonal matrix

J = (\begin{matrix}b_{1} & a_{1} & 0 & 0 & … \\ a_{1} & b_{2} & a_{2} & 0 & … \\ 0 & a_{2} & b_{3} & a_{3} & … \\ 0 & 0 & a_{3} & b_{4} & … \\ ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix})

viewed as an operator on ℓ^{2}(ℤ_{+}) via

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

Here a_{n} > 0 and b_{n} ∈ ℝ. We will sometimes denote the variables in J explicitly by writing J({a_{n}}_{n≥1}, {b_{n}}_{n≥1}). We are interested in perturbations of the special case a_{n} ≡ 1, b_{n} = 0, called J_{0}, the free Jacobi matrix and, in particular, the case where J − J_{0} is compact, viz. a_{n} → 1, b_{n} → 0 as n → ∞. Then σ_{ess}(J) = σ_{ess}(J_{0}) = [−2, 2] and J has simple eigenvalues {E_{n}^{±}}_{n=0}^{N_{±}} with (N_{+} or N_{−} or both may be infinite)

E_{1}^{+} > E_{2}^{+} > ⋯ > 2 > −2 > ⋯ > E_{2}^{−} > E_{1}^{−}.  (1.2)

One of our main goals in this paper is to prove the bound

Theorem 1.

∑_{n=1,...,N_{±}}[(E_{n}^{+})^{2} − 4]^{1/2} + [(E_{n}^{−})^{2} − 4]^{1/2} ≤ ∑_{n}|b_{n}| + 4 ∑_{n}|a_{n} − 1| (1.3)

Date: November 30, 2001.

1 Department of Mathematics 253–37, California Institute of Technology, Pasadena, CA 91125, U.S.A.; E-mail: dirkh@caltech.edu, bsimon@caltech.edu.

2 Supported in part by NSF grant DMS-9707661.

As we will see, the constants 1 in front of the b sum and 4 in the a_{n} − 1 sum are both optimal. (1.3) is optimal in another regime, namely, large coupling for b. Specifically, let J_{λ} be defined with a_{n} = a_{n}^{(0)} and b_{n} = λb_{n}^{(0)}. Let b̃_{n}^{±} be a reordering of the b_{n}’s with ±b̃_{n}^{±} > 0 so b̃_{1}^{+} ≥ b̃_{2}^{+} ≥ ⋯ ≥ 0 and b̃_{1}^{−} ≤ b̃_{2}^{−} ≤ ⋯ ≤ 0. Then it is not hard to see that

lim_{λ→∞} λ^{−1}E_{n}^{±}(J_{λ}) = b̃_{n}^{±}  (1.4)

which shows that the ratio of the two sides of (1.3) goes to 1 as λ → ∞ for any b_{n} with ∑|b_{n}| < ∞.

Since

(E_{n}^{±})^{2} − 4 = |E_{n}^{±} ∓ 2| |E_{n}^{±} ± 2|
≥ 4|E_{n}^{±} ∓ 2|,

(1.3) implies that

∑_{n}|E_{n}^{+} − 2|^{1/2} + |E_{n}^{−} + 2|^{1/2} ≤ \frac{1}{2}(∑_{n}|b_{n}| + 4 ∑_{n}|a_{n} − 1|).  (1.5)

More generally, we will prove that

Theorem 2.

∑_{n}|E_{n}^{+} − 2|^{p} + |E_{n}^{−} + 2|^{p} ≤ c_{p}[∑_{n}|b_{n}|^{p+1/2} + 4 ∑_{n}|a_{n} − 1|^{p+1/2}] (1.6)

for any p ≥ \frac{1}{2} where

c_{p} = \frac{1}{2} 3^{p−1/2} \frac{Γ(p + 1)}{Γ(p + \frac{3}{2})} \frac{Γ(2)}{Γ(\frac{3}{2})} .

As for sums of moments for p < \frac{1}{2}, we will prove

Theorem 3. Let 0 ≤ p < \frac{1}{2}. Let ∥ · ∥ be any translation invariant norm on pairs of sequences {a_{n}}_{n=0}^{∞}, {b_{n}}_{n=0}^{∞}. For any ε > 0, there exists a Jacobi matrix with a_{n} = 1, b_{n} = 0 for n large so that ∥(a, b)∥ ≤ ε but ∑_{n}|E_{n}^{+} − 2|^{p} + |E_{n}^{−} − 2|^{p} ≥ ε^{−1}.

As (1.4) shows, (1.5) and (1.6) are poor as λ → ∞, since the left side grows like λ^{p} and the right side as λ^{p+1/2}. It is better to use

(E_{n}^{±})^{2} − 4 = |E_{n}^{±} − 2| |E_{n}^{±} ± 2|
≥ |E_{n}^{±} − 2|^{2}

and (1.3) to obtain

∑_{n}|E_{n}^{+} − 2| + |E_{n}^{−} + 2| ≤ ∑_{n}|b_{n}| + 4 ∑_{n}|a_{n} − 1|  (1.7)

and the related Theorem 4.

∑_{n}|E_{n}^{+} − 2|^{p} + |E_{n}^{−} + 2|^{p} ≤ ∑_{n}(b_{n}^{+} + 2|a_{n} − 1|)^{p} + (b_{n}^{−} + 2|a_{n} − 1|)^{p} (1.8)

As (1.4) shows, the ratio of the two sides of (1.8) is 1 as λ → ∞.

We got interested in this problem because Killip-Simon [14] needed a bound like Theorem 1 to prove a conjecture of Nevai [20, 21] that if the right side of (1.3) is finite, then a condition of Szegö holds. They and we expected bounds like (1.3) to hold because of the analogous results for Schrödinger operators.

Nevai’s conjecture says that if ∑_{n}|b_{n}| + ∑_{n}|a_{n} − 1| < ∞, then, with m, the m-function defined by

m(E) = (J − E)_{11}^{−1},

we have

∫_{−2}^{2} log Im m(E + i0) \frac{dE}{\sqrt{4 − E^{2}}} > −∞.  (1.9)

Killip-Simon [14] use a sum rule of Case [4, 5] that

Z(m) = ∑_{n} log|a_{n}| + ∑ log|β_{j}|  (1.10)

where β_{j} is defined by |β_{j}| > 1 and β_{j} +β_{j}^{−1} are the listing of the eigenvalues of J outside [−2, 2]. In (1.10), Z(m) is defined by

Z(m) = \frac{1}{2π} ∫_{−2}^{2} log(\frac{\sqrt{4 − E^{2}}}{Im m(E + i0)})\frac{dE}{\sqrt{4 − E^{2}}} .  (1.11)

(1.10) is only proven initially for J with J − J_{0} finite rank. (Or, in any event, not initially for all J’s with J − J_{0} trace class. Eventually, using our bounds here and the theory of Nevanlinna functions, Killip-Simon [14] do prove (1.10) for trace class J − J_{0}.) Killip-Simon show Z(m) is lower semicontinuous as a trace class J is approximated by cutoff J’s with J − J_{0} finite rank. Thus to prove Z(m) < ∞ (i.e., that (1.9) holds), they need to control the right side of (1.10). Since ∑|a_{n} − 1| < ∞, the ∑_{n} log|a_{n}| is absolutely convergent. Since |β_{j}| ∼ 1 + (|E_{j}| − 2)^{1/2} for E_{j} close to 2, (1.5) implies that ∑ log|β_{j}| is uniformly bounded.

Theorem 1 should also be interesting in connection with some recent results of Peherstorfer-Yuditskii [22], who focus on the finiteness of the right side of (1.3).

Bounds for Schrödinger operator eigenvalues of the form

∑_{n=1}^{∞}|E_{n}|^{p} ≤ L_{p,ν} ∫_{ℝ^{ν}}|V (x)|^{p+ν/2} d^{ν}x  (1.12)

where E_{n} are the negative eigenvalues of −Δ+V on L^{2}(ℝ^{ν}) go back twentyfive years to the work of Lieb and Thirring [18, 19], who used the case p = 1, ν = 3 in their celebrated proof of the stability of matter. They proved (1.12) for p > 0, ν ≥ 2, and p > \frac{1}{2}, ν = 1, and shortly thereafter, Cwikel [7], Lieb [17], and Rozenblum [24] proved (1.12) in case p = 0, ν ≥ 3. It is easy to see (e.g., Landau-Lifshitz [15, pp. 156–157] and Simon [26]) that it is false in case p = 0, ν = 2.

For many years, the case p = \frac{1}{2}, ν = 1 was open, perhaps in part because [19] erroneously claimed to have proven it. Only in 1996 did Weidl [27] establish this result for p = \frac{1}{2}, ν = 1. For ν = 1, Lieb-Thirring [19] conjectured the optimal value of L_{p,ν} for all p ≥ \frac{1}{2}. They proved their conjecture when ν = 1 for p = \frac{3}{2}, \frac{5}{2}, \frac{7}{2}, …, and subsequently, Aizenman-Lieb [1] for all p ≥ \frac{3}{2}. Shortly after Weidl’s work, Hundertmark, Lieb, and Thomas [12] found a new proof which yielded the optimal constant L_{1/2,1}. A partially alternate proof of a part of the argument in [12] can be found in Hundertmark, Laptev, and Weidl [11].

Unlike the discrete case, the continuum theory has a scaling symmetry: taking V (x) → λ^{2}V (λx) yields E_{n} → λ^{2}E_{n} since there is a unitary operator that implements x → λx. This forces the power |E|^{p} on the right side of (1.12) given the scaling behavior of d^{ν}x. Thus the same power properly captures large and small E’s. In the discrete case, this is not so, which is why we have two bounds (1.5) and (1.8). As noted, (1.8) is good for large coupling, but (1.5) is better for small E’s. In particular, if b_{n} ∼ n^{−α} (with α > 1) for n large, (1.8) only implies ∑|E_{n}^{+} − 2|^{p} < ∞ for p > α^{−1} which (1.5) implies is true for p > α^{−1} − \frac{1}{2}.

Of course, the best extended estimate would involve powers of (E^{2} − 4)^{1/2} but both the Aizenman-Lieb [1] method to increase powers and the Laptev-Weidl [16] method to increase dimension seem to require powers of dist(E, σ_{ess}(J)). However, one can save a little bit of the structure; see the remark at the end of section 5.

We note one interesting feature of (1.3) vis-à-vis the continuum bound. The continuum p = \frac{1}{2} bound has an optimal constant, but is off by a factor of 2 in the large coupling limit. For (1.3), as we noted above, the optimal bound for small coupling is also exact in the large coupling limit.

In Section 2, we will prove Theorem 1 when a_{n} ≡ 1 by closely following [12] and then obtain Theorems 2 and 4 when a_{n} ≡ 1 by the now standard argument of Aizenman and Lieb [1]. In Section 3, we make a simple but useful observation that allows one to obtain estimates for eigenvalues for arbitrary Jacobi matrices from the estimates for the special case. Section 4 contains some examples and some counterexamples, and proves Theorem 3. Section 5 uses ideas of Laptev-Weidl [16] to prove bounds for the higherdimensional case. In an appendix, we show how the ideas in this paper provide a simple proof of a strengthening of the Bargmann-type bound of Geronimo [8, 9].

This paper is aimed towards two rather different audiences: the Schrödinger operator community and the orthogonal polynomial community, who have rather different toolkits. For that reason, we include some material (such as that at the start of Section 2) that one group or the other may regard as elementary.

Acknowledgment. We thank Jeff Geronimo, Fritz Gesztesy, Rowan Killip, and Paul Nevai for useful comments.

2. Bounds for Discrete Schrödinger Operators

In this section, we prove Theorems 1, 2, and 4 when all a_{n} = 1. We begin with some general preliminaries. Given any self-adjoint operator A, bounded from above, we define

E_{j}^{+} = inf_{φ_{1}...φ_{j−1}} sup_{\begin{matrix}ψ:ψ⊥φ_{j} \\ ψ∈D, ∥ψ∥=1 \end{matrix}} ⟨ψ, Aψ⟩.  (2.1)

Similarly, if A is bounded below,

E_{j}^{−} = sup_{φ_{i}...φ_{j}} inf_{\begin{matrix}ψ:ψ⊥φ_{j} \\ ψ∈D(A), ∥ψ∥=1 \end{matrix}} ⟨ψ, Aψ⟩  (2.2)

We will use E_{j}^{±}(A) if the dependence on A is important. From the definitions,

A ≤ B ⇒ E_{j}^{±}(A) ≤ E_{j}^{±}(B)  (2.3)

and

E_{1}^{−} ≤ E_{2}^{−} ≤ ⋯ ≤ E_{2}^{+} ≤ E_{1}^{+}.  (2.4)

The min-max principle (Theorem XIII.1 in Reed-Simon [23]) asserts that (i) E_{∞}^{±} = lim E_{j}^{±} has E_{∞}^{+}(A) = sup σ_{ess}(A), E_{∞}^{−}(A) = inf σ_{ess}(A) (ii) If A has N^{+} (resp. N^{−}) eigenvalues counting multiplicity in the

interval (E_{∞}^{+}, ∞) (resp. (−∞, E_{∞}^{−})), these eigenvalues are precisely E_{1}^{±}, E_{2}^{±}, …, E_{N_{±}}^{±} and E_{j}^{±} = E_{∞}^{±} if j > N_{±}.

Next, note from the definition that if A_{m} → A in norm, then we have convergence of the corresponding eigenvalues since |E_{j}^{±}(A)−E_{j}^{±}(B)| ≤ ∥A− B∥. It follows if f is an arbitrary continuous nonnegative function, then

∑_{j=1}^{k} f(E_{j}^{±}(A)) = lim_{m→∞} ∑_{j=1}^{k} f(E_{j}^{±}(A_{m}))

≤ lim inf_{m→∞} ∑_{j=1}^{∞} f(E_{j}^{±}(A_{m}))

so taking k → ∞,

∑_{j=1}^{∞} f(E_{j}^{+}(A)) + f(E_{j}^{−}(A)) ≤ lim inf_{m→∞} ∑_{j=1}^{∞} f(E_{j}^{+}(A_{m})) + f(E_{j}^{−}(A_{m})). (2.5)

(2.5) and the min-max principle imply Proposition 2.1. To prove (1.3)–(1.6), it suffices to prove the special case where only finitely many a_{n}’s differ from 1 and finitely many b’s differ from 0.

Next, we want to note the impact of restriction. Let A be a bounded self-adjoint operator on H. Let P be an orthogonal projection. By A_{P}, we mean P AP restricted as an operator on P H = Ran P. In (2.1)/(2.2), changing from A to A_{P} adds the condition ψ ∈ Ran P and it decreases sups and increases infs. Thus

Proposition 2.2.

E_{j}^{+}(A_{P}) ≤ E_{j}^{+}(A);  E_{j}^{−}(A_{P}) ≥ E_{j}^{−}(A)  (2.6)

We have two applications of (2.6) in mind. First, given two two-sided sequences {a_{n}}_{n=−∞}^{∞}, {b_{n}}_{n=−∞}^{∞}, define the whole-line operator W on ℓ^{2}(ℝ) by

(W u)(n) = a_{n−1}u(n − 1) + b_{n}u(n) + a_{n}u(n + 1).  (2.7)

Thus, if P is the projection of ℓ^{2}(ℤ) to ℓ^{2}(ℤ_{+}) ⊂ ℓ^{2}(ℤ), W_{P} = J where J is built from the projected sequences {a_{n}}_{n=1}^{∞} and {b_{n}}_{n=1}^{∞}. As a result, (2.6) implies

Proposition 2.3. To prove (1.3)–(1.6), it suffices to prove the analogous result for the whole-line operators.

One might think that the results are much harder for whole-line operators. After all, it can be shown that if b̃ has compact support, then J(a_{n} ≡ 1, b_{n} = λb̃_{n}) has no spectrum outside [−2, 2] if λ is small, but W (a_{n} ≡ 1, b_{n} = λb̃_{n}) always has eigenvalues outside [−2, 2] if λ ≠ 0, b̃ ≢ 0. That is why there is a Bargmann bound for J but not for W. However, it is not harder because (1.3)–(1.6) have translation invariant quantities for their right side. Let P_{n} be the projection onto ℓ^{2} (m ∈ ℤ, m ≥ n). One can see that as n → −∞, E_{j}^{±}(W_{P_{n}}) → E_{j}^{±}(W) so (1.3)–(1.6) for the Jacobi case actually implies it for the whole-line case.

The second application of (2.6) is to the study of the following objects that will play a role below:

S_{n}^{±}(A) = ∑_{j=1}^{n} E_{j}^{±}(A).  (2.8)

Proposition 2.4. Let A be a self-adjoint operator.

(i) S_{n}^{+}(A) = sup{Tr(AP) | P^{∗} = P, P^{2} = P, Tr(P) = n} (ii) S_{n}^{−}(A) = inf{Tr(AP) | P^{∗} = P, P^{2} = P, Tr(P) = n} (iii) A ↦ S_{n}^{+}(A) is convex; A ↦ S_{n}^{−}(A) is concave.

Remark. One can see that if E_{∞}^{+} ≥ 0, then in (i) P^{2} = P can be replaced by ∥P ∥ ≤ 1 which is how it is often written.

Proof. (i) By (2.6), S_{n}^{+}(A_{P}) ≤ S_{n}^{+}(A). But since Ran P has dimension n, S_{n}^{+}(A_{P}) = Tr(A_{P}) = Tr(AP). Thus

S_{n}^{+}(A) ≥ sup{Tr(AP) | P^{∗} = P, P^{2} = P, Tr(P) = n}.

Next, pick φ_{1}, …, φ_{n} as follows. If n ≤ N^{+}(A), take φ_{1}, …, φ_{n} to be the eigenfunctions of A with eigenvalues E_{1}^{+}, …, E_{n}^{+}. If n > N^{+}, pick φ_{1}, …, φ_{N^{+}} to be the eigenfunctions for A with eigenvalues E_{1}^{+}, …, E_{N^{+}}^{+} and φ_{N^{+}+1}, …, φ_{n} to be arbitrary orthonormal vectors in Ran(P_{[E_{∞}^{+}−ε,E_{∞}^{+}]}(A)), the range of the spectral projection which is infinite-dimensional when N^{+} < ∞ since E_{∞}^{+} = sup σ_{ess}(A).

Let P be the projection onto the span of φ_{1}, …, φ_{n}. Then

Tr(AP) = ∑_{j=1}^{n}(φ_{j}, Aφ_{j})
≥ S_{n}^{+}(A) − ε[min(n, N^{+}) − n].

Since ε is arbitrary,

S_{n}^{+}(A) ≤ sup{Tr(AP) | P^{∗} = P, P^{2} = P, Tr(P) = n}.

(ii) The same proof as (i).

(iii) S_{n}^{±} are the sup and inf of linear functions, so convex and concave, respectively. □

As a final general preliminary, we note the Birman-Schwinger principle: Let A be a self-adjoint operator which is bounded above with α = sup σ(A). Let B be a positive relatively form compact, that is,

K_{β} ≡ B^{1/2}(β − A)^{−1}B^{1/2}  (2.9)

is compact for one and hence for all β > α. K_{β} is called the Birman-Schwinger operator.

Proposition 2.5 (The Birman-Schwinger Principle [3, 25]). Let λ > 0. β > α is an eigenvalue of A + λB if and only if K_{β} has eigenvalue λ^{−1}. We have for j ≤ N^{+}(A + λB),

E_{j}^{+}(K_{E_{j}^{+}(A+λB)}) = λ^{−1}.  (2.10)

Remark. The point of (2.10) is that the index j is the same in both E_{j}^{+}’s.

Proof. For simplicity, we suppose A and B are bounded operators, which is true in the applications we will make. If (A + λB)φ = βφ, then B^{1/2}(β − A)^{−1}B^{1/2}(B^{1/2}φ) = λ^{−1}B^{1/2}φ and B^{1/2}φ ≠ 0 since if not, we must have Aφ = βφ, which is impossible since β > sup σ(A). Conversely, if K_{β}ψ = λ^{−1}ψ and φ = (β − A)^{−1}B^{1/2}ψ (≠ 0 since λ^{−1} ≠ 0), we have (A + λB)φ = βφ. Thus the first expression is true.

Next, note that ∥K_{β}∥ → 0 as β → ∞ by compactness. Its eigenvalues are continuous, and so by eigenvalue perturbation theory [13, 23], real analytic.

If e(β) is a positive eigenvalue of K_{β} with K_{β}φ = eφ and ∥φ∥ = 1, then by eigenvalue perturbation theory (the Feynman-Hellmann theorem),

\frac{de}{dβ} = ⟨φ, \frac{∂K_{β}}{∂β} φ⟩ = −∥(β − A)^{−1}B^{1/2}φ∥^{2} < 0,

so e is strictly monotone. Thus if e(β) is the jth eigenvalue of K_{β} and e(β_{0}) > λ^{−1}, there is exactly one β > β_{0} with e(β) = λ^{−1}, so

#{j | E_{j}^{+}(K_{β_{0}}) ≥ λ^{−1}} = #{β > β_{0} | E_{j}^{+}(K_{β}) = λ^{−1}}

(counting multiplicity) from which (2.10) follows. □

With the general preliminaries out of the way, we compute the Birman-Schwinger operator for A = W_{0} and a diagonal (i.e., a_{n} ≡ 1) perturbation.

Proposition 2.6. Let W_{0} be the whole-line matrix with a_{n} ≡ 1, b_{n} ≡ 0. Let β > 2 = sup σ(W_{0}). Then (β − W_{0})^{−1} has matrix elements

[(β − W_{0})^{−1}]_{nm} = (μ^{−1} − μ)^{−1}μ^{|n−m|}  (2.11)

where μ is related to β by

β = μ + μ^{−1};  μ < 1.  (2.12)

Remark. Of course, μ = \frac{1}{2}(β − \sqrt{β^{2} − 4} ) and μ^{−1} = \frac{1}{2}(β + \sqrt{β^{2} − 4} ) so μ^{−1} − μ = \sqrt{β^{2} − 4}. This is why \sqrt{E^{2} − 4} enters in Theorem 1.

Proof. This is a standard calculation. Looking for solutions of

φ(n − 1) + φ(n + 1) = βφ(n),  (2.13)

one tries φ(n) = ζ^{n} and finds ζ + ζ^{−1} = β, so the solutions are ζ = μ and ζ = μ^{−1}. Let

φ_{±}(n) = μ^{±n}.

Both solve (2.13) if μ obeys (2.12). Since μ < 1, φ_{+} is ℓ^{2} at +∞, φ_{−} at −∞, so the right side of (2.11) which has the form (μ^{−1}−μ)^{−1}φ_{−}(min(n, m))φ_{+}(max(n, m)) ≡ G_{n}(m) is ℓ^{2} in m for each n with ((W_{0} − β)G_{n})(m) = 0 if m ≠ n. By a direct computation (essentially μ^{−1} − μ is the Wronskian of φ_{+} and φ_{−}), (β_{0} − W)G_{n} = δ_{n}, that is, G_{n}(m) = [(β_{0} − W)^{−1}δ_{n}](m), proving (2.11). □

Remark. Alternatively, one can use Fourier analysis to compute the inverse.

Because of (2.11), the following operator will enter in our discussion, {b_{n}}_{n∈ℤ} is a positive sequence of finite support,

(L_{μ})_{nm} = b_{n}^{1/2}μ^{|n−m|}b_{m}^{1/2}.  (2.14)

Recall the definition (2.8) of S_{m}( ·). The crucial lemma is

Proposition 2.7. Let 0 < μ < η ≤ 1. Then for any n,

S_{n}^{+}(L_{μ}) ≤ S_{n}^{+}(L_{η}).  (2.15)

Remarks. 1. Since Tr(L_{μ}) is constant, individual eigenvalues cannot all be monotone.

2. This is a special case of the warm-up to the proof of Lemma 4 in [12]. Our proof is close to the proof there, except where [12] uses eigenvalue perturbation at μ_{j} = 0, we use symmetry.

Proof. Given a bounded positive sequence {μ_{n}}_{n=−∞}^{∞}, we define

(L_{{μ_{n}}})_{kℓ} = b_{k}^{1/2}b_{ℓ}^{1/2} ∏_{j=k}^{ℓ−1} μ_{j}  if k ≤ ℓ
= (L_{{μ_{n}}})_{kℓ}  if k > ℓ

so L_{μ} is L_{{μ_{n}}} when all μ_{n} = μ. Thus (2.15) follows if we show S_{n}^{+}(L_{{μ_{n}}}) is monotone in μ_{n} ∈ [0, ∞) when {μ_{j}}_{j≠n} are held fixed. Let f(μ) be this function when μ_{n} takes the value μ. L_{{μ_{j}}_{j≠n}, μ_{n}=μ} is affine in μ for each matrix element is either constant or a multiple of μ. More precisely, in matrix notation we have

L_{\{μ_{j}\}_{j≠n},μ_{n}=μ} = (\begin{matrix}A & 0 \\ 0 & B \end{matrix}) + μ (\begin{matrix}0 & C \\ C^{†} & 0 \end{matrix})

where A, B, and C depend only on {μ_{j}}_{j≠n}. So by Proposition 2.4 (iii), f(μ) is a convex function of μ.

On the other hand, if U is the diagonal matrix,

(Uφ)(ℓ) = −φ(ℓ)  ℓ ≤ n
= φ(ℓ)  ℓ ≥ n + 1,

or, as a block matrix, U = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ), then UL_{\{μ\}}U^{−1} = L_{\{μ̃\}} where

μ̃_{ℓ} = μ_{ℓ}  if ℓ ≠ n
= −μ_{ℓ}  if ℓ = n,

that is, we have

U (\begin{matrix}A & μC \\ μC^{†} & B \end{matrix}) U^{−1} = (\begin{matrix}A & −μC \\ −μC^{†} & B \end{matrix}) = L_{\{μ_{j}\}_{j≠n}, μ_{n}=−μ}.

Since E_{j}^{+}, and so S_{j}^{+}, are invariant under unitary transformations, we see f(−μ) = f(μ). An even convex function is monotone increasing on [0, ∞), so S_{n}^{+}(L_{{μ_{n}}}) is monotone in each μ_{n} in the region μ_{n} ≥ 0. □

We are now ready to prove what is essentially Theorem 1 in case a_{n} ≡ 1.

Theorem 2.8. Let W_{0} be the free whole-line Schrödinger operator and B a positive finite-rank diagonal matrix. Let W = W_{0} + B. Then

∑_{j=1}^{N_{+}(W)} \sqrt{E_{j}^{+}(W)^{2} − 4} ≤ Tr(B).  (2.16)

Proof. (Following [12]) Since B is finite rank, we know that N_{j}^{+}(W) < ∞. Define μ_{j} by μ_{j}^{−1} + μ_{j} = E_{j}^{+} with μ_{j} < 1. By (2.9) and the remark after Proposition 2.6,

K_{E_{j}^{+}} = ((E_{j}^{+})^{2} − 4)^{−1/2}L_{μ_{j}}  (2.17)

with L_{μ} given by (2.14). By (2.10),

E_{j}^{+}(K_{E_{j}^{+}}) = 1.  (2.18)

Since for a > 0, E_{j}^{+}(aA) = aE_{j}^{+}(A), (2.17), (2.18) imply

\sqrt{E_{j}^{+}(W)^{2} − 4} = E_{j}^{+}(L_{μ_{j}}).  (2.19)

Thus

∑_{j=1}^{N^{+}(W)} \sqrt{E_{j}^{+}(W)^{2} − 4} = E_{1}^{+}(L_{μ_{1}}) + E_{2}^{+}(L_{μ_{2}}) + ⋯ + E_{N^{+}}^{+}(L_{μ_{N^{+}}} ). (2.20)

But, by (2.15) and μ_{1} < μ_{2} < ⋯ < μ_{N_{+}} < 1,

E_{1}^{+}(L_{μ_{1}}) + E_{2}^{+}(L_{μ_{2}}) = S_{1}^{+}(L_{μ_{1}}) + E_{2}^{+}(L_{μ_{2}})
≤ S_{1}^{+}(L_{μ_{2}}) + E_{2}^{+}(L_{μ})
= S_{2}^{+}(L_{μ_{2}})
≤ S_{2}^{+}(L_{μ_{3}}),

so by induction,

∑_{j=1}^{k} E_{j}^{+}(L_{μ_{j}}) ≤ S_{k}^{+}(L_{μ_{k}}) ≤ S_{k}^{+}(L_{μ_{k+1}})

and thus (2.20) implies

∑_{j=1}^{N^{+}(W)} \sqrt{E_{j}^{+}(W)^{2} − 4} ≤ ∑_{j=1}^{N^{+}} E_{j}^{+}(L_{μ=1}) = Tr(B)

since L_{μ=1} is the rank one operator b_{n}^{1/2}b_{m}^{1/2} with a single nonzero eigenvalue equal to Tr(L_{μ=1}) = Tr(B). □

Remark. The proof shows the inequality is strict if E_{1}^{+}(L_{μ}) is strictly monotone. Thus the inequality is strict if rank (B) ≥ 2.

There is a standard argument of Aizenman-Lieb [1] which we can use to go from a (\frac{1}{2}, 1) bound (power of E − 2, power of b) to a general (p, p + \frac{1}{2}) for any p ≥ \frac{1}{2}:

Theorem 2.9. Under the hypothesis of Theorem 2.8, we have, for any p ≥ \frac{1}{2},

∑_{j=1}^{N_{+}(W)}|E_{j}^{+}(W) − 2|^{p} ≤ \frac{1}{2} \frac{Γ(p + 1)}{Γ(p + \frac{3}{2})} \frac{Γ(2)}{Γ(\frac{3}{2})} Tr(|B|^{p+1/2}).  (2.21)

Proof. Note first that since b_{n} ≤ (b_{n})_{+} ≡ max(0, b_{n}), if positivity of B is dropped, we still have that

∑_{j=1}^{N_{+}(W)}|E_{j}^{+}(W) − 2|^{1/2} ≤ \frac{1}{2} ∑_{n}(b_{n})_{+}  (2.22)

by using (2.5), W_{0} + B ≤ W_{0} + B_{+} and \sqrt{E^{2} − 4} ≤ 2|E − 2|^{1/2}.

Let r > 0. Then

(E_{j}^{+}(W) − 2 − r)_{+} = (E_{j}^{+}(W − r1) − 2)_{+}

so (2.22) implies

∑_{j=1}^{N_{+}(W)}|E_{j}^{+}(W) − 2 − r|_{+}^{1/2} ≤ \frac{1}{2} ∑_{n}(b_{n} − r)_{+}.  (2.23)

Now the well-known integral for α < p,

\frac{Γ(p + 1)}{Γ(p − α)Γ(α + 1)} ∫_{0}^{1}(1 − x)^{α}x^{p−α−1} dx = 1

with scaling implies for any α < p:

a_{+}^{p} = C_{p,α} ∫_{0}^{∞}(a − r)_{+}^{α}r^{p−α−1} dr  (2.24)

where C_{p,α} = Γ(p + 1)/Γ(p − α)Γ(α + 1). (2.23) and (2.24) immediately imply that

∑_{j=1}^{N_{+}(W)}|E_{j}^{+}(W) − 2|^{p} ≤ \frac{1}{2} \frac{C_{p,1/2}}{C_{p+1/2,1}} ∑_{n}(b_{n})_{+}^{p+1/2}  (2.25)

which implies (2.21). □

Similarly, we have

Theorem 2.10. Under the hypothesis of Theorem 2.8, we have for any p ≥ 1,

∑_{j=1}^{N_{+}(W)}|E_{j}(W) − 2|^{p} ≤ Tr(|B|^{p}).  (2.26)

Proof. Since E ≥ 2 implies

E^{2} − 4 = (E − 2)^{2}(E + 2)
≥ (E − 2)^{2},

(2.16) implies (2.26) for p = 1. The result for general p ≥ 1 follows as above. Where above we get a factor of C_{p,1/2}/C_{p−1/2,1}, here we get C_{p,1}/C_{p,1} = 1. □

So far, we have proven a bound on E_{j}^{+}, but they immediately imply bounds on E_{j}^{−}. One can prove that by analogy, but it is even easier to use the unitary map

(V u)(n) = (−1)^{n}u(n)

which has

V W ({a_{n}}, {b_{n}})V^{−1} = W ({−a_{n}}, {b_{n}}) = −W ({a_{n}}, {−b_{n}}),

so

E_{j}^{−}(W ({a_{n}}, {b_{n}})) = −E_{j}^{+}(W ({a_{n}}, {−b_{n}})).  (2.27)

Thus, for example,

∑_{j=1}^{N_{r}}|E_{j}^{−}(W)^{2} − 4|^{1/2} ≤ ∑_{n}(−b_{n})_{+}
= ∑_{n}(b_{n})_{−}  (2.28)

where x_{−} = (−x)_{+} = − min(0, x) so |x| = x_{+} + x_{−} and we obtain (1.3) for the case a_{n} ≡ 1.

3. Bounds for Jacobi Matrices

The following elementary observation lets us pass from bounds in case a_{n} ≡ 1 to the general case. Note that

(\begin{matrix}−|a_{n} − 1| & 1 \\ 1 & −|a_{n} − 1| \end{matrix}) ≤ (\begin{matrix}0 & a_{n} \\ a_{n} & 0 \end{matrix}) ≤ (\begin{matrix}|a_{n} − 1| & 1 \\ 1 & |a_{n} − 1| \end{matrix})

for any a_{n} real since for any x in ℝ, ( \begin{matrix}|x| & x \\ x & |x| \end{matrix} ) ≥ 0 since it has determinant 0 and trace 2|x| ≥ 0. This immediately implies by repeated use at each pair of indices

W ({a_{n} ≡ 1}, {b_{n}^{−}}) ≤ W ({a_{n}}, {b_{n}}) ≤ W ({a_{n} ≡ 1}, {b_{n}^{+}})  (3.1)

where

b_{n}^{±} = b_{n} ± (|a_{n−1} − 1| + |a_{n} − 1|).  (3.2)

(3.1) and (2.5) immediately imply Theorem 3.1. Let f be monotone increasing on (0, ∞) and even. Then

f(E_{j}^{±}(W ({a_{n}}, {b_{n}}))) ≤ f(E_{j}^{±}(W ({a_{n} ≡ 1}, {b_{n}^{±}})))  (3.3)

where b_{n}^{±} is given by (3.2).

With this, we can now prove our three main theorems:

Proof of Theorem 1. By (3.3), (2.23), and (2.28),

∑_{n}[(E_{n}^{+})^{2} − 4]^{1/2} + [(E_{n}^{−})^{2} − 4]^{1/2}

≤ ∑_{n}[b_{n} + |a_{n−1} − 1| + |a_{n} − 1|]_{+} + [b_{n} − |a_{n−1} − 1| − |a_{n} − 1|]_{−}

≤ ∑_{n}[b_{n}]_{+} + [b_{n}]_{−} + 4 ∑_{n}|a_{n} − 1|.  (3.4)

In obtaining (3.4), we used [x + y]_{+} ≤ x_{+} + y_{+} and [x + y]_{−} ≤ x_{−} + y_{−} and that a given |a_{n} − 1| occurs in four terms with [b_{n}]_{±} and [b_{n+1}]_{±}. □

Proof of Theorem 2. By (3.3), (2.21), and (2.27),

\begin{matrix}∑_{n} & |E_{n}^{+} − 2|^{p} + ∑_{n}|E_{n}^{−} + 2|^{p} ≤ \\ d_{p} ∑_{n}[b_{n} + |a_{n−1} − 1| + |a_{n} − 1|]_{+}^{p+1/2} + [b_{n} − |a_{n−1} − 1| − |a_{n} − 1|]_{−}^{p+1/2} \end{matrix} ; (3.5)

where

d_{p} = \frac{1}{2} \frac{Γ(p + 1)}{Γ(p + \frac{3}{2})} \frac{Γ(2)}{Γ(\frac{3}{2})} .

Now for any q ≥ 1 (q will be p + \frac{1}{2}), x^{q} is convex, so

(α + β + γ)^{q} = 3^{q}(\frac{α}{3} + \frac{β}{3} + \frac{γ}{3} )^{q}
≤ 3^{q−1}[α^{q} + β^{q} + γ^{q}]

from which (1.6) holds if we note that c_{p} = 3^{(p+1/2)−1} d_{p}. □

Proof of Theorem 4. As stated, (1.8) is an immediate consequence of Theorem 2.10 and (3.1).

We kept this bound in the form (1.8) to get an exact result as λ → ∞. We could use the same method of proof of Theorem 2 to get

∑_{n}|E_{n}^{+} − 2|^{p} + ∑_{n}|E_{n}^{−} + 2|^{p} ≤ 3^{p−1}[∑_{n}|b_{n}|^{p} + 4 ∑_{n}|a_{n} − 1|^{p}].

□

We believe it could be true that (1.3) holds with |a_{n} − 1| replaced by (a_{n} − 1)_{+} and, in particular, we know that (1.6) and (1.8) hold when p ≥ 1 if |a_{n} − 1| is replaced by (a_{n} − 1)_{+}. To see the latter, we note that — by a convexity plus evenness argument much like that in the proof of Proposition 2.7 — ∑_{j=1}^{k} E_{j}^{+}(W ({a_{n}}, {b_{n}})) is monotone in a_{n} in the region a_{n} ≥ 0. Thus for p = 1, (1.6) and (1.8) hold with (a_{n}−1)_{+} for we move those a’s with a_{n} > 1 to the diagonal as we did in (3.1), and use the monotonicity just noted to move a_{n}’s in (0, 1) up to 1. Once one has the result for p = 1, it follows for p ≥ 1 by the Aizenman-Lieb argument.

The fact that in (1.6) for p ≥ 1 and in the Bargmann bound of the Appendix, one can take (a_{n} − 1)_{+} leads us to conjecture (1.3) holds with (a_{n} − 1)_{+} rather than |a_{n} − 1|.

4. Examples

Example 4.1. W has all a_{n} = 1, all n, and all b_{n} = 0 for n ≠ 0. If b_{0} ≡ b > 0, then there is an eigenvalue at energy E = μ + μ^{−1} with μ < 1 and eigenfunction φ_{n} = μ^{|n|}. To have the eigenfunction fit at n = 0, we need

2μ + b1 = E1

or

b = μ^{−1} − μ = \sqrt{E^{2} − 4} .

This example has equality in (1.3) for all values of b > 0 (and also b < 0 it turns out) and shows one cannot decrease the value 1 in front on ∑|b_{n}|. □

Example 4.2. W has all b_{n} = 0, all n, and all a_{n} = 1, n ≠ 0. If a_{0} ≡ a > 1, there is an eigenvalue at energy E = μ + μ^{−1} with 0 < μ < 1. Then φ_{n} = μ^{−n} for n ≤ 0 and φ_{n} = μ^{n−1} for n ≥ 1 since φ must be symmetric around n = \frac{1}{2}. The eigenfunction condition at 0 reads

μ + a = μ + μ^{−1}

or a = μ^{−1}. Thus

a − a^{−1} = \sqrt{E^{2} − 4} .

There is a second eigenvalue at energy −E (there has to be by the symmetry (2.27)). Thus

LHS of (1.3) = 2(a − a^{−1})
= 2(1 + a^{−1})(a − 1).

The two sides of (1.3) are not equal for any a, but the ratio goes to 1 as a ↓ 1 since 2(1 + a^{−1}) ↑ 4. Thus the 4 in front of the |a − 1| cannot be made smaller. However, both this example and the discussion in the appendix suggest it might be possible to replace |a − 1| by (a − 1)_{+}. □

As noted above, the best constant for the W case is the same as for the J case.

Example 4.3 (Proof of Theorem 3). Shift to the Jacobi case. Take an example with a_{n} ≡ 1 and b_{n} = 0, except for n = m, 2m, …, Nm where b_{n} = β. As m → ∞, there are n eigenvalues above 2 which all approach the solution of \sqrt{E^{2} − 4} = β. So long as β < 1, |E_{n} − 2| ≥ \frac{1}{6}β^{2}, so

∑_{n}|E_{n} − 2|^{p} ≥ N(\frac{β^{2}}{6} )^{p}.  (4.1)

In the translation invariant norm ∥ · ∥, let α = ∥(a_{n} ≡ 1, b_{1} = 1, b_{n} = 0 for n ≠ 1)∥. Then for the (a, b) of this β, N, m example,

∥(a, b)∥ ≤ Nαβ.  (4.2)

Let N_{0}(ε), β_{0}(ε) solve

N(\frac{β}{6} )^{p} = 2ε^{−1}
Nαβ = \frac{ε}{2}

so

β = c_{1}ε^{2/1−2p} → 0
N = c_{2}ε^{−(1+2p)/(1−2p)} → ∞

since p < \frac{1}{2}. Increase N slightly to be an integer. Thus

∑_{n}|E_{n} − 2|^{p} ≥ ε^{−1}, ∥(a, b)∥ ≤ ε,

proving Theorem 3. □

5. Bounds in Higher Dimension

In this section, we want to use the ideas of Laptev-Weidl [16] to prove bounds on operators on ℓ^{2}(ℤ^{ν}). We begin with the discrete Schrödinger operator case. Let H_{0} be defined on ℓ^{2}(ℤ^{ν}) by

(H_{0}u)(n) = ∑_{|m−n|=1} u(m)

and

(V u)(n) = V (n)u(n).

Lemma 5.1. Let W_{0} act on ℓ^{2}(ℤ; X) where X is a Hilbert space, and let B(n) : X → X be self-adjoint and trace class with ∑_{n} Tr(|B(n)|) < ∞. Then

∑_{j}(E_{j}^{±}(W_{0} + B)^{2} − 4)^{1/2} ≤ Tr_{X}(B^{±})  (5.1)

where B^{±}(n) = max(±B(n), 0) is defined via the functional calculus.

Proof. Suppose B(n) ≥ 0. As with (2.14), define L_{μ} : ℓ^{2}(ℤ; X) → ℓ^{2}(ℤ; X) by

(L_{μ})_{mn} = B_{n}^{1/2}μ^{|n−m|}B_{m}^{1/2}.

As with Proposition 2.7, 0 < μ < η ≤ 1 implies

S_{n}^{+}(L_{μ}) ≤ S_{n}^{+}(L_{η})

and then the proof of (2.16) extends. □

Theorem 5.2. If V ∈ L^{p}(ℤ^{ν}; X) for p ≥ 1 where X is a Hilbert space, that is, V (x) : X → X is a symmetric compact operator such that ∑_{x∈ℤ^{ν}} Tr_{X}|V (x)|^{p} < ∞, then

∑_{j}|E_{j}^{+}(H_{0} + V) − 2ν|^{p} + ∑_{j}|E_{j}^{−}(H_{0} + V) + 2ν|^{p} ≤ ∑_{x∈ℤ^{ν}} Tr_{X}|V (x)|^{p}. (5.2)

Proof. By the Aizenman-Lieb idea, (2.24), it suffices to prove this for p = 1. As usual, we can suppose V ≥ 0 and prove the result for E_{j}^{+}. Write

H_{0} = H_{0,1} + H_{0,{2,...,ν}}

where H_{0,1} involves neighbors in the 1 direction and H_{0,{2,...,ν}} neighbors in the other directions. Note that

(H_{0,1}+H_{0,{2,...,ν}}+V −2ν)_{+} ≤ (H_{0,1}+(H_{0,{2,...,ν}}+V −2(ν−1))_{+}−2)_{+} (5.3)

and thus

∑_{j}|E_{j}^{+}(H_{0} + V) − 2ν| = Tr_{ℓ^{2}(ℤ^{ν};X)}((H_{0} + V − 2ν)_{+})

≤ Tr_{ℓ^{2}(ℤ;ℓ^{2}(ℤ^{ν−1};X))}((H_{0,1} + (H_{0,{2,...,ν}} + V − 2(ν − 1))_{+} − 2)_{+})

≤ ∑_{n_{1}} Tr_{ℓ^{2}(ℤ^{ν−1};X)}((H_{0,{2,··· ,ν}} + V (n_{1}, ·) − 2ν + 2)_{+})

by (5.1) and (E^{2} − 4)^{1/2} ≥ (|E| − 2). An inductive argument completes the proof. □

For the other moment result, it will be convenient to phrase things in terms of the classical constants,

L_{p,ν}^{cℓ} = (2π)^{−ν/2} ∫_{|k|≤1}|k|^{2p} d^{ν}k
= 2^{−ν}π^{−ν/2} \frac{Γ(p + 1)}{Γ(p + 1 + \frac{ν}{2} )} .  (5.4)

These constants have several important features. First, the argument that led to (2.25) says that if

∑_{j=1}^{N_{+}}|E_{j}^{+} − 2|^{p} ≤ αL_{p,ν}^{cℓ} ∑_{n}|b_{n}|^{p+ν/2}  (5.5)

for some p = p_{0}, it holds for all p > p_{0}. Second,

L_{p=1/2, ν=1} = 2^{−1}π^{−1/2}[ \frac{\sqrt[\frac{1}{2}]{π}}{1} ] = \frac{1}{4}

so the consequence of (5.1) and (E^{2} − 4)^{1/2} ≥ 2(|E| + 2)^{1/2} is that (5.5) holds for ν = 1, p = \frac{1}{2}, and α = 2.

Finally, we note that from (5.4) and Fubini, we have

L_{p,ν}^{cℓ} = ∏_{j=0}^{ν=1} L_{p+j/2,1}^{cℓ}.  (5.6)

Theorem 5.3. Let V ∈ L^{p+ν/2}(ℤ^{ν}; X) for p ≥ 1. Then

∑_{j}|E_{j}^{+}(H_{0}+V)−2ν|^{p}+∑_{j}|E_{j}^{−}(H_{0}+V)+2ν|^{p} ≤ 2^{ν}L_{p,ν}^{cℓ} ∑_{x∈ℤ^{ν}} Tr_{X}|V (x)|^{p+ν/2}.

(5.7)

Proof. We exploit (5.3), but use (5.5) for α = 2, ν = 1, p ≥ \frac{1}{2} at each stage of the induction. We then get (5.7) with a constant

∏_{j=0}^{ν−1} 2L_{p+1/2,1}^{cℓ} = 2^{ν}L_{p,ν}^{cℓ}

by (5.6). □

As in the one-dimensional case, Theorem 5.2 is better for large coupling. Indeed, it is exact in the large coupling regime, while Theorem 5.3 gives more information on the eigenvalues very close to ±2ν in the regime of slow decay of V (n) at infinity.

As with the one-dimensional case, we can handle nonconstant off-diagonal terms which approach 1 fast enough at infinity. Let B(ℤ^{ν}) be set of bonds in ℤ^{ν}, that is, the set of unordered pairs b = (ij) with i, j ∈ ℤ^{ν}, |i − j| = 1. Given {a_{b}}_{b∈B(ℤ^{ν})}, a nonnegative real number a_{b} for each bond b = (ij), one can define

(H_{0}(a_{b})u)(n) = ∑_{|m−n|=1} a_{(nm)}u(m).  (5.8)

The analog of (3.3) is then

H_{0} + V^{−} ≤ H_{0}(a_{b}) + V ≤ H_{0} + V^{+}  (5.9)

where

V^{±}(n) = V (n) ± ∑_{|m−n|=1}|a_{(mn)} − 1|  (5.10)

so, for example, we get

∑_{j}[E_{j}^{+}(H_{0}(a_{b}) + V) + E_{j}^{−}(H_{0}(a_{b}) + V)] ≤ ∑_{n}|V (n)| + ∑_{b} 4|a_{b} − 1|. (5.11)

Remark. Since the bound in Theorem 1 is optimal both for large and small coupling, the curious reader might wonder whether it is possible to keep some of its structure also in higher dimension. This is indeed the case. For constant diagonal terms and scalar potential we have the two bounds

∑_{n=1,...,N_{±}}[(E_{n}^{+})^{2} − 4]^{1/2} + [(E_{n}^{−})^{2} − 4]^{1/2} ≤ ∑_{x∈ℤ^{ν}}|V (x)|

and

∑_{n=1,...,N_{±}}[(E_{n}^{+})^{2} − 4]^{1/2} + [(E_{n}^{−})^{2} − 4]^{1/2} ≤ 2^{ν−1}L_{1,ν−1}^{cℓ} ∑_{x∈ℤ^{ν}}|V (x)|^{1+(ν−1)/2}.

Simply use the induction in the dimension idea to strip off the first coordinate x_{1} and then use either Theorem 5.2 or 5.3 in ν − 1 dimension. Of course, the above extension to nonconstant diagonal terms also applies.

Appendix A. The Bargmann Bound

Our goal in this appendix is to prove

Theorem A.1. Let N({a}, {b}) be the number of eigenvalues of J({a}, {b}) outside [−2, 2]. Then

N({a}, {b}) ≤ ∑_{n=1}^{∞} n|b_{n}| + (4n + 2)(a_{n} − 1)_{+}  (A.1)

where (x)_{+} = max(x, 0).

This is related to a result of Geronimo [8, 9]. We provide a proof here because it is easy from our machinery earlier. Geronimo’s second proof of this result [9] uses a Birman-Schwinger kernel as this does, but has an error in the argument that allows a_{n} < 1 (his Lemma III.1 is wrong). Earlier papers that show N < ∞ if (A.1) holds include Geronimo-Case [10] and Chihara-Nevai [6].

Notes. 1. If you translate Geronimo’s result in [8] into our normalization (he has J_{0} with a ≡ \frac{1}{2}, not a = 1), then where we have (4n + 2)(a_{n} − 1)_{+}, he has (4n+4)(a_{n}−1)_{+}(a_{n}+1), which is weaker in two regards: 4n+2 < 4n+4 and we have no a_{n} + 1. We note that by looking at b_{n} = 0 and a_{n} = 1 for n ≥ 2, one finds examples with N = 2 and (a_{1} − 1)_{+} arbitrarily close to \sqrt{2} − 1 so that constant in front of (a_{1} − 1)_{+} must be at least 2(\sqrt{2} + 1) and, in particular, 4n does not work.

2. We actually have separate inequalities for N_{+} and N_{−}.

Step 1. a_{n} ≡ 1; b_{n} ≥ 0. The proof of Bargmann’s bound [2] given by Birman [3] and Schwinger [25] works in this case. By (2.10) and the monotonicity with A = J_{0}, B = J − J_{0}, for β > 2,

# of eigenvalues of A + B ≥ β

= # of β′ ≥ β so that K_{β′} has eigenvalue = 1

= # of eigenvalues of K_{β} ≥ 1  (A.2)

≤ Tr(K_{β})  (A.3)
≤ Tr(K_{2})  (A.4)

where (A.2) follows from the fact that ∥K_{β}∥ ↓ 0 as β → ∞ and the strict monotonicity of the eigenvalues of K_{β} noted in the proof of Proposition 2.5. (A.3) holds since

Tr(K_{β}) = ∑_{E_{j}^{+}(K_{β})} E_{j}^{+} ≥ ∑_{E_{j}^{+}(K_{β})≥1} E_{j}^{+} ≥ (# of eigenvalues of K_{β} ≥ 1)

since K_{β} > 0. (A.4) holds since K_{β} ≤ K_{2}.

The same argument that led to Proposition 2.6 shows that [(β−J_{0})^{−1}]_{nm} = w(β)^{−1}φ_{−}^{(β)}(min(n, m))φ_{+}^{(β)}(max(n, m)) where φ_{±} solve J_{0}φ = βψ with φ_{+} ∈ L^{2} at infinity, φ_{−}(0) = 0, and w is their Wronskian. As β ↓ 2, φ_{+}(n) → 1, φ_{−}(n) → n, and their Wronskian is 1 so

(K_{2})_{nm} = min(n, m)b_{n}^{1/2}b_{m}^{1/2}

and

Tr(K_{2}) = ∑_{n=1}^{∞} nb_{n},

proving (A.1) in this case.

Step 2. a_{n} ≤ 1; b_{n} ≥ 0. Let J_{0}({a_{n}}) be J with b_{n} = 0. We claim if a_{n} ≤ 1 and β > 2, then

(β − J_{0}({a_{n}}))_{nm}^{−1} ≤ (β − J_{0})_{nm}^{−1}.  (A.5)

This is a simple maximal principle argument. One first notes that if f_{n}(m) = (β −J_{0}({a_{n}}))_{nm}^{−1}, then f_{n}(m) > 0 (expand (1−β^{−1}J) in a geometric series). Next, one notes that

((β − J_{0})f_{n})(m) = δ_{nm} − (1 − a_{m−1})f_{n}(m − 1) − (1 − a_{m})f_{n}(m)
≤ δ_{nm}.

Since (β − J_{0})^{−1} also has a positive matrix, applying it preserves pointwise matrix inequalities, so

f_{n}(m) ≤ [(β − J_{0})^{−1}δ_{n}]_{m} = (β − J_{0})_{nm}^{−1},

proving (A.5).

Now (A.5) shows the Birman-Schwinger kernel for J_{0}{a_{n}} and J({a_{n}, b_{n}}) is dominated (in the sense of inequalities on matrix elements) by this for J_{0} and J({a_{n} ≡ 1, b_{n}}), so Step 1 implies

Tr(K_{2}({a_{n}, b_{n}})) ≤ Tr(K_{2}({a_{n} ≡ 1, b_{n}})) = ∑_{j=1}^{∞} nb_{n}.

Notice we do not have an operator inequality of the form (β−J_{0}({a_{n}}))^{−1} ≤ (β − J_{0})^{−1}, so individual eigenvalues may not have an inequality (this is Geronimo’s error in [9]).

Step 3. Adding b’s of both signs. Fix a_{n} with 0 < a_{n} ≤ 1. Let J({b_{n}}) be the Jacobi matrix with b_{n} along the diagonal and N_{±}({b_{n}}) the number of eigenvalues E with ±E > 2. Since J({−(b_{n})_{−}}) ≤ J({b_{n}}) ≤ J({(b_{n})_{+}}), we have

N_{±}({b_{n}}) ≤ N_{±}({±(b_{n})_{±}}),

so by (A.1) for b_{n} ≥ 0 and (2.27), we have (A.1) for the case 0 ≤ a_{n} ≤ 1.

Step 4. (General Case) Now use the idea at the start of Section 3 but only for a_{n}’s with a_{n} > 1. Then (A.1) holds in general since this idea reduces to the case a_{n} ≤ 1. We use here that

2n(a_{n} − 1)_{+} + 2(n + 1)(a_{n} − 1)_{+} = (4n + 2)(a_{n} − 1)_{+}.

References

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[3] M.S. Birman, The spectrum of singular boundary problems, Mat. Sb. (N.S.) 55 (97) (1961), 125–174 (Russian). Translated in Amer. Math. Soc. Trans. 53 (1966), 23–80.

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