Abstract
The research on spectral inequalities for discrete Schrödinger 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 regard to connecting higher order Schrödinger-type operators with symmetric matrix operators with arbitrarily many nonzero diagonals above and below the main diagonal. We thus obtain spectral bounds for such matrices, similar in nature to the Lieb-Thirring inequalities.
1. Background
Let be the self-adjoint Jacobi matrix operator acting on as follows: via where and . This operator can be viewed as the one-dimensional discrete Schrödinger operator if 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.
If , rapidly enough, as , the essential spectrum of is absolutely continuous and coincides with the interval (see, e.g., [3]). Besides, may have simple eigenvalues where , and Indeed, in [2] the authors found the following.
Theorem 1. If , , , then where
The author (see [4]) then improved their result, achieving the smaller constant: , by translating a well-known 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 for higher order Schrödinger-type operators and thus “polydiagonal” Jacobi-type matrix operators, which we will define below.
2. Notation and Preliminary Material
For a sequence , let and be the difference operator and its adjoint, respectively, denoted by and . We then denote the discrete one-dimensional Laplacian by . For , , and a sequence , with , we define by We note that being self-adjoint immediately implies that is also self-adjoint.
Finding an explicit formula for requires a few combinatorial techniques, all of which are standard. Let , for . Then we have the following: (i) , (ii) , and (iii) .
A simple induction argument then delivers our formula for the σth 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 will thus be the range of the above symbol, which can be found to be .
3. Main Results
We now let , , be the orthonormal system of eigensequences in corresponding to the negative eigenvalues of the ()th order discrete Schrödinger-type operator as follows: where and we assume that for all . Our next result is concerned with estimating those negative eigenvalues.
Theorem 2. Let , , . Then the negative eigenvalues of the operator satisfy the inequality where
Remark 3. As the discrete spectrum of lies in and , we shift our operator to the left by and by analogy have an estimate for the positive eigenvalues of that operator, thus immediately obtaining Corollary 4.
Corollary 4. Let , , . Then the positive eigenvalues of the operator satisfy the inequality:
Finally we will apply these results to obtain spectral bounds for the following operator.
We let be a polydiagonal self-adjoint Jacobi-type matrix operator as follows: viewed as an operator acting on as follows: for , , where , , for all . We denote where we understand to mean . We are then interested in perturbations of the following special case: where , and explicitly called the free Jacobi-type matrix of order . In particular, we examine the case where is compact. Thus in what follows we assume that our sequences tend to the operator coefficients rapidly enough; that is, , , as . Then the essential spectrum is given by and may have simple eigenvalues where , and
Theorem 5. Let , , and for all . Then for the eigenvalues of the operator we have where
4. Auxiliary Results
We require the following discrete Kolmogorov-type inequality.
Lemma 6. For a sequence , and for , we have the following inequality:
Proof. We proceed by induction, where we note that the initial case, , , 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 . 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 . Thus We thus apply our induction hypothesis and set and as follows: We now return to the induction hypothesis as follows:
We are now equipped to prove an Agmon-Kolmogorov-type inequality.
Proposition 7. For a sequence , we have for any
Proof. First we use Lemma 6 with , as follows: and we apply this estimate to the well-known discrete Agmon inequality (see [4]):
Proposition 8. Let be an orthonormal system of sequences in ; that is, , and let . Then
Proof. Let . By Proposition 7, we have Let and as ,
5. 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 be the negative eigenvalues of the operator . By the variational principle for the negative eigenvalues of the operator we have By this estimate, we find that by (38) above, where is the well-known Beta function. Thus, after a change of variable, completing our proof.
6. Proof of Theorem 5
We have the following matrix bounds for square, matrices, as given in [2]. For , , we have We thus use this on each block of indices of as follows: where is given by that is, Now we relate these to our Schrödinger-type operators: Now () are positive eigenvalues of . Thus by using (43) and the variational principle, we have where are the positive eigenvalues of Let us now define , . Then, by Corollary 4 for the positive eigenvalues of our operator, we have Thus, applying (48), where Similarly, using Theorem 2 on (47), Using the following application of Jensen’s inequality, that is, for , let , with , to each of (51) and (53), we have Summing these two inequalities, we arrive at where and the proof of Theorem 5 is complete.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.