Resonances on regular tree graphs

TL;DR

This paper studies the distribution of resonances near spectral thresholds for Laplace operators on regular tree graphs with branching, showing results on resonance absence and discrete spectrum near certain spectral sectors.

Contribution

It provides new insights into the resonance distribution and spectral properties of Laplace operators on regular tree graphs with nonself-adjoint perturbations.

Findings

Resonances are absent near spectral thresholds under certain conditions.

Discrete spectrum is absent near specific sectors of the essential spectrum.

Results apply to nonself-adjoint exponentially decaying potentials.

Abstract

We investigate the distribution of the resonances near spectral thresholds of Laplace operators on regular tree graphs with $k$-fold branching, $k \geq 1$, perturbed by nonself-adjoint exponentially decaying potentials. We establish results on the absence of resonances which in particular involve absence of discrete spectrum near some sectors of the essential spectrum of the operators.