On the Number of Discrete Eigenvalues of a Discrete Schrödinger Operator with a Finitely Supported Potential

Abstract

On the d-dimensional lattice Z d and the r-regular tree T r , an exact expression for the number of discrete eigenvalues of a discrete Laplacian with a finitely supported potential is described in terms of the support and the intensities of the potential on each case. In particular, the number of eigenvalues less than the infimum of the essential spectrum is bounded by the number of negative intensities.Mathematics Subject Classifications. 39A12, 47A75, 34L40.

Weighted estimates for the Laplacian on the cubic lattice

“…Note that in [5] the authors gave an example in dimension d 5 of an embedded eigenvalue at the endpoint {±d}. Recently, Hayashi, Higuchi, Nomura, and Ogurisu computed the number of discrete eigenvalues for finitely supported potential [4].…”

“…Note that for n = 0, the estimate (A.2) implies (A.6). We estimate first supposing |t − n| n 1/3 : t 1 4 (n…”