Counting eigenvalues of Schr\"odinger operator with complex fast decreasing potential

TL;DR

This paper provides sharp estimates for the number of eigenvalues of discrete Schr"odinger operators with rapidly decreasing complex potentials, extending to non-symmetric Jacobi matrices, using advanced complex analysis techniques.

Contribution

It introduces a novel method to estimate eigenvalues for complex, rapidly decreasing potentials in Schr"odinger operators and Jacobi matrices.

Findings

Sharp eigenvalue estimates for complex potentials

Application to non-symmetric Jacobi matrices

Enhanced understanding of eigenvalue distribution

Abstract

We give a sharp estimate of the number of zeros of analytic functions in the unit disc belonging to analytic quasianalytic Carleman--Gevrey classes. As an application, we estimate the number of the eigenvalues for discrete Schr\"odinger operators with rapidly decreasing complex-valued potentials, and, more generally, for non-symmetric Jacobi matrices.