A remark on the absence of eigenvalues in continuous spectra for discrete Schr\"{o}dinger operators on periodic lattices

TL;DR

This paper proves a theorem regarding the absence of eigenvalues in the continuous spectrum for Schrödinger operators with decaying potentials on various periodic lattices, enhancing understanding of spectral properties.

Contribution

It establishes a Rellich-Vekua type theorem and discusses unique continuation and non-existence of embedded eigenvalues for these operators on multiple lattice types.

Findings

Eigenvalues are absent in the continuous spectrum for the studied operators.

The paper extends spectral theory results to various periodic lattice structures.

It confirms the non-existence of embedded eigenvalues in this context.

Abstract

We prove a Rellich-Vekua type theorem for Schr\"{o}dinger operators with exponentially decreasing potentials on a class of lattices including square, triangular, hexagonal lattices and their ladders. We also discuss the unique continuation theorem and the non-existence of eigenvalues embedded in the continuous spectrum.