Schr\"odinger operators with guided potentials on periodic graphs

TL;DR

This paper studies how guided potentials on periodic graphs influence the spectrum of discrete Schr"odinger operators, revealing the emergence and properties of guided spectral bands and their dependence on graph geometry.

Contribution

It introduces a detailed analysis of the guided spectrum for Schr"odinger operators with periodic and localized potentials on graphs, including estimates and asymptotics.

Findings

Guided spectrum consists of finitely many bands added to the unperturbed spectrum.

Guided bands' positions and lengths depend on graph geometric parameters.

The number and properties of guided bands can be arbitrarily varied with specific potentials.

Abstract

We consider discrete Schr\"odinger operators with periodic potentials on periodic graphs perturbed by guided non-positive potentials, which are periodic in some directions and finitely supported in other ones. The spectrum of the unperturbed operator is a union of a finite number of non-degenerate bands and eigenvalues of infinite multiplicity. We show that the spectrum of the perturbed operator consists of the unperturbed one plus the additional guided spectrum, which is a union of a finite number of bands. We estimate the position of the guided bands and their length in terms of graph geometric parameters. We also determine the asymptotics of the guided bands for large guided potentials. Moreover, we show that the possible number of the guided bands, their length and position can be rather arbitrary for some specific potentials.