Spectral Bounds for Polydiagonal Jacobi Matrix Operators

The research on spectral inequalities for discrete Schrodinger Operators has proved fruitful in the last decade. Indeed, several authors analysed the operator's canonical relation to a tridiagonal Jacobi matrix operator. In this paper, we consider a generalisation of this relation with regards to connecting higher order Schrodinger-type operators with symmetric matrix operators with arbitrarily many non-zero diagonals above and below the main diagonal. We thus obtain spectral bounds for such matrices, similar in nature to the Lieb{Thirring inequalities.


Background
Let be the self-adjoint Jacobi matrix operator acting on ℓ 2 (Z) as follows: via ( ) ( ) = −1 ( − 1) + ( ) + ( + 1) , where > 0 and ∈ R. This operator can be viewed as the one-dimensional discrete Schrödinger operator if = 1 for all . A variety of papers examined such operators; for example, we quote the work by Killip and Simon in [1], where they obtained sum rules for such Jacobi matrices. Additionally, Hundertmark and Simon in [2] were able to find spectral bounds for these operators. We thus state their result.
The author (see [4]) then improved their result, achieving the smaller constant: = 3 −1 ,1 , by translating a wellknown method employed by Dolbeaut et al. in [5] to the discrete scenario. They, in turn, used a simple argument by Eden and Foias (see [6]) to obtain improved constants for Lieb-Thirring inequalities in one dimension.
The aim of this paper is to answer the natural question of whether these methods can be generalised to give bounds 2 International Journal of Partial Differential Equations for higher order Schrödinger-type operators and thus "polydiagonal" Jacobi-type matrix operators, which we will define below.
Finding an explicit formula for Δ requires a few combinatorial techniques, all of which are standard. Let := ( ) := !/(( − )! !), for , ∈ N. Then we have the following: . A simple induction argument then delivers our formula for the th order discrete Laplacian operator as follows: Furthermore, in order to identify our essential spectrum, we apply the discrete fourier transform as follows: which, after some rearrangement, yields The essential spectrum of the operator Δ will thus be the range of the above symbol, which can be found to be

Main Results
We now let { } =1 , ∈ N, be the orthonormal system of eigensequences in ℓ 2 (Z) corresponding to the negative eigenvalues { } =1 of the (2 )th order discrete Schrödingertype operator as follows: where ∈ {1, . . . , } and we assume that ≥ 0 for all ∈ Z. Our next result is concerned with estimating those negative eigenvalues.

Auxiliary Results
We require the following discrete Kolmogorov-type inequality.
Lemma 6. For a sequence ∈ ℓ 2 (Z), and for > ≥ 1, we have the following inequality: Proof. We proceed by induction, where we note that the initial case, = 1, = 2, holds true as the inequality is in fact the simple inequality found by Copson in [7]. This case in turn, if used repeatedly, shows that the inequality holds true for all , if = +1. We then take the inductive step on the variable . Hence we assume that we have the required inequality for < ≤ , given a fixed , and proceed to prove the statement for = + 1. Thus We thus apply our induction hypothesis and set = − 1 and = as follows: We now return to the induction hypothesis as follows: We are now equipped to prove an Agmon-Kolmogorovtype inequality.

Proof of Theorem 2
We take the inner product with ( ) on (10) and sum both sides of the equation with respect to . We obtain We now use Proposition 8 and apply the appropriate Hölder's inequality; that is, We define .
The latter inequality can be written as The LHS is maximal when Substituting this into (33), we obtain Therefore, We lift this bound now with regard to moments by using the standard Aizenman-Lieb procedure (see [8]). We let By this estimate, we find that completing our proof.
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Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.