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Eigenvalue Bounds for Perturbations of Schrodinger Operators and Jacobi Matrices With Regular Ground States

We prove general comparison theorems for eigenvalues of perturbed Schrodinger operators that allow proof of Lieb--Thirring bounds for suitable non-free Schrodinger operators and Jacobi matrices.


Introduction
Consider a Schrödinger operator, on L 2 (R ν ) which we suppose obeys inf spec(−∆ + V 0 ) = 0 (1.2) (By subtracting a constant, we can always arrange this, and by assuming this, the notation simplifies.) We are interested in controlling the negative eigenvalues of (1.4) be either the negative eigenvalues or 0, that is, We say that H 0 has a regular ground state if and only if there exists a function, u 0 , on R ν obeying for some c 1 , c 2 . We take c 1 = inf u, c 2 = sup u, and let Our main result is that any bound on the number or sums of eigenvalues for the operator −∆ + V can be carried (with a change in the coupling constant) to the operator H 0 + V. This is based on the following observation: Theorem 1.1. For any V ≤ 0 and V 0 obeying (1.2) with regular ground state, we have for all j, This result is remarkable for its generality and also for the simplicity of its proof. We will see in Section 2 that it can be used to compare not only V 0 and 0 but two arbitrary V 0 's with relatively bounded ground states.
Of course, (1.9) immediately implies bounds on moments of bound states: where we look at this only for γ ≥ 0 and interpret 0 0 = 0 so S 0 is the number of strictly negative eigenvalues. Clearly, Theorem 1.1 implies: For any γ > 0 and β = β(V 0 ) given by (1.8), The standard Lieb-Thirring inequalities (reviewed in [8,14]) assert for γ ≥ 1 2 in ν = 1, γ > 0 in ν = 2, and γ ≥ 0 in ν ≥ 3. In some cases, the optimal constants are known, for example, L1 2 ,1 = 1 2 . (These yield good constants in our perturbed estimates but we do not claim optimal constants for our situation!) Clearly, (1.11) implies: and, in particular, One can also obtain logarithmic estimates as in [13] and Hardy-Lieb-Thirring bounds as in [5]. Since it is known [20,17,14] that for ν = 1 and V ≤ 0, These results are of interest because there are many cases which are known to have regular ground states. Example 1.4. If V 0 is periodic, then there is a positive periodic ground state. If V 0 is locally L ν 2 (if ν ≥ 3, locally L 1 if ν = 1, and locally L p with p > 1 if ν = 2), then it is known that eigenfunctions are continuous (see [18]) and thus, H 0 has a regular ground state. Example 1.5. We will discuss Jacobi matrices in Sections 3 and 4. It is known (see [19,15,3]) that elements in the isospectral torus of finite gap Jacobi matrices have regular ground states. Example 1.6. If u is any function obeying (1.7), then V 0 = ∆u u has a regular ground state.
In Section 2, we will review the ground state representation and prove a stronger theorem than Theorem 1.1. As hinted, it is the ground state representation that is critical. In this regard, we should emphasize that the variational argument we use in Section 2 has appeared earlier in work of the Birman school-we would mention, in particular, Lemma 6.1 of Birman, Laptev, and Suslina [2], although it may have appeared earlier in their work. Our novelty here is the wide applicability, the use in CLR and Lieb-Thirring bounds, and the applicability to the discrete case and Szegő estimates.
As we will explain in Section 4, an initial motivation for this work was critical Lieb-Thirring bounds for finite gap almost periodic Jacobi matrices in connection with Szegő's theorem for such situations.
Ground state representations do not seem to be in the literature for Jacobi matrices, so we do this first in Section 3, and then prove an analog of Theorem 1.1 for Jacobi matrices in Section 4. Section 5 discusses some other cases.
It is a pleasure to thank Fritz Gesztesy, Yehuda Pinchover, Robert Seiringer, and Simone Warzel for useful comments, and Michael Aizenman for being a sensitive editor.

Comparison for Schrödinger Operators
Fundamental to our results is the ground state representation that if (1.6) holds for u 0 , continuous and strictly positive on R ν , then Ground state representations go back to Jacobi [11]. For Schrödinger operators, it appears at least as far back as Birman [1] and it was used extensively in constructive quantum field theory (especially by Segal, Nelson, Gross, and Glimm-Jaffe; see Glimm-Jaffe [7]). As a basis for comparison theorems, it was used by Kirsch-Simon [12] and, as noted above, in a similar context by Birman-Laptev-Suslina [2].
We will be cavalier about technical assumptions needed for (2.1). From one point of view, we can use (2.1) as a definition of H 0 ! Namely, the right side of (2.1) defined for g's with distributional derivative making the right side finite is easily seen to be a closed quadratic form on Notice that we do not need (1.2), but only inf spec(H 0 ) ≥ 0 for this to work, and Theorem 2.1 below holds in that case (although inf spec(H 0 ) = inf spec(H 1 )) under the hypothesis of the theorem. Here is our main result: For any V ≤ 0, let E j be given by (1.5). Then Remark. By interchanging V 0 and V 1 and replacing V by β −1 V, we get the complementary inequality Proof. Let Since −V g 2 ≥ 0 and |∇g| 2 ≥ 0, we have so, by (2.10) and (2.1), and τ > 0, so RHS of (2.13) ≤ −τ gu 1 , gu 1 proving (2.7).
Proof of Theorem 2.1.
Thus, there is a space of dimension at least j where (2.16) holds. By the min-max principle (see [16]), which is (2.4).

Ground State Representation for Jacobi Matrices
While we are interested mainly in semi-infinite one dimension Jacobi matrices, that is, tridiagonal semi-infinite matrices, we can consider the higher-dimensional case as well, so we will. So far as we know, there is no prior literature on the ground state representation for discrete operators, so we start with that in this section.
In Z ν , we let δ j , j = 1, . . . , ν, be the ν component vectors with 1 in the jth place and 0 elsewhere. So k ±δ j are the 2ν neighbors of k ∈ Z ν . A Jacobi operator is parametrized by a symmetric a jℓ > 0 for each j, ℓ ∈ Z ν with |j − ℓ| = 1 and a real number b k for each k ∈ Z ν . We will suppose sup k |b k | + sup ℓ |a kℓ | < ∞ The Jacobi operator associated with these parameters is the operator J on ℓ 2 (Z ν ) with We will use J({a ℓm }, {b ℓ }) if we want to make the dependence on a and b explicit.
for all ℓ ∈ Z ν . Then for any f with f u ∈ ℓ 2 (Z ν ), we have Remark. In particular, this shows −J ≥ 0.
Proof. It suffices to prove (3.5) for f of finite support and take limits.
For the left side converges since −J is a bounded operator, and by positivity, the right side converges (a priori perhaps to ∞, but by the equality to a finite limit; it is only here that positivity of u is used).

Comparison for Jacobi Operators
In this section, we will prove an analog of Theorem 1.1 for Jacobi operators. One difference that we have to expect is that of sup spec(J) = 0; the same is true of λJ for any λ > 0, but the ground states are the same. Thus, comparison of the two J's cannot involve only the ground state ratio but also a setting of scales which will enter as a ratio of a's. In the Schrödinger case, the scale is set by the −∆ rather than −λ∆.
For notation, we let E j ({a ℓm , b ℓ }) be the max of zero and the jth eigenvalue of J({a ℓm , b ℓ }) counting from the top. Here is our main result: ℓ } be two sets of bounded Jacobi parameters with positive sequences u (0) , u (1) obeying (3.4) for Suppose β + < ∞ and β − > 0. Then for perturbations {δa ℓm , δb ℓ } with a (0) Remarks. 1. We only have a one-sided inequality as we would in the Schrödinger case if we did not demand V ≤ 0. Since δa terms are never of a definite sign, we cannot have them in a two-sided comparison. But there is clearly a two-sided comparison if δa = 0 and δb > 0. 2. Note that rescaling u (0) or u (1) which changes β + , β − , γ − does not change η or β. Similarly, η scales properly under changes of the scale of a.
As in [9], this follows from 0 a a 0 ≤ a 0 0 a and b ≤ |b|.
Proof of Theorem 4.1. The proof is identical to the proof in Section 2, the sole change being that (2.11) needs to be replaced by with |n| = |n 1 | + · · · + |n ν |. Then This allows one to control eigenvalues below inf spec(J) in the same way. We define E − j to be the min of 0 and the jth eigenvalue of J counting from the bottom. We call a sequence u on Z ν alternating positive if (−1) |m| u m > 0 for all m. The result is: ℓ } be two sets of bounded Jacobi parameters with alternating positive sequences u ℓ } be a one dimension periodic set of Jacobi parameters or the almost periodic parameters associated with a finite gap spectrum (see [19]). Let J 0 be the associated half-line Jacobi matrix. Let J be the Jacobi matrix associated with {a Let E − 1 < E − 2 < · · · < inf spec(J 0 ) < sup spec(J 0 ) < · · · < E + 2 < E + 1 be the eigenvalues of J outside the convex hull of spec(J 0 ). Then for a constant C depending only on J 0 .
Proof. We only need that J 0 and WJ 0 W −1 have regular ground states. This follows from Floquet theory for the periodic case and from the detailed analysis of Jost solutions for the almost periodic case; see [19,15,3]. Then compare to the free J 0 (a ℓm = 1 if |ℓ − m| = 1, b ℓ = 0) and use the bound of [9].
For the periodic case, this is proven by Damanik, Killip, and Simon [4], who also prove this where the sum in (4.11) is over all eigenvalues including the ones in gaps.
It remains an interesting question relevant to the study of the Szegő condition (see [10,3]) to get the bound in the almost periodic case. In [10], Hundertmark and Simon prove the weaker bounds where the 1 power is replaced by any p > 1 2 or where p = 1 2 but there is an ℓ ε added to the sum.

Some Final Remarks
All we needed for our arguments is some kind of ground state representation. That means we can replace −∆ by and we can still compare to −∆ although sup A and inf A will enter. We can compare magnetic field operators where the magnetic field is fixed but For the discrete case, the key was not tridiagonal matrices, but ones with a ground state representation. For example, non-negative offdiagonal will do.