Location of eigenvalues of non-self-adjoint discrete Dirac operators

TL;DR

This paper provides quantitative bounds on the eigenvalues of one-dimensional discrete Dirac operators with complex potentials, identifying regions free of embedded eigenvalues and discussing the bounds' sharpness.

Contribution

It introduces new spectral bounds for discrete Dirac operators with complex potentials, including criteria for absence of embedded eigenvalues in the essential spectrum.

Findings

Eigenvalue bounds for complex $ ext{ extlangle} p ext{ extrangle}$-potentials

Identification of spectrum regions free of embedded eigenvalues for small $ ext{ extlangle} 1 ext{ extrangle}$-potentials

Discussion on the sharpness and potential improvements of spectral bounds

Abstract

We provide quantitative estimates on the location of eigenvalues of one-dimensional discrete Dirac operators with complex $\ell^p$-potentials for $1\leq p\leq\infty$. As a corollary, subsets of the essential spectrum free of embedded eigenvalues are determined for small $\ell^1$-potential. Further possible improvements and sharpness of the obtained spectral bounds are also discussed.