# Spectral enclosures for non-self-adjoint discrete Schr\"odinger operators

## TL;DR

This paper investigates the eigenvalue locations of one-dimensional discrete Schrödinger operators with complex potentials in various ℓ^p spaces, establishing optimal bounds for ℓ^1 and new bounds for p>1 using the Birman-Schwinger method.

## Contribution

It provides the first optimal eigenvalue bounds for ℓ^1 potentials and introduces new spectral bounds for ℓ^p potentials with p>1 in discrete Schrödinger operators.

## Key findings

- Optimal eigenvalue bounds for ℓ^1 potentials.
- New spectral bounds for p>1 potentials.
- Method relies on Birman-Schwinger principle and norm estimations.

## Abstract

We study location of eigenvalues of one-dimensional discrete Schr\"odinger operators with complex $\ell^{p}$-potentials for $1\leq p\leq \infty$. In the case of $\ell^{1}$-potentials, the derived bound is shown to be optimal. For $p>1$, two different spectral bounds are obtained. The method relies on the Birman-Schwinger principle and various techniques for estimations of the norm of the Birman-Schwinger operator.