Periodic Solutions of Generalized Schr\"odinger Equations on Cayley Trees

TL;DR

This paper investigates periodic solutions of generalized fractional Schr"odinger equations on Cayley trees, introducing a discrete fractional Laplacian and providing criteria for the existence of periodic solutions based on subgroup properties.

Contribution

It defines a discrete generalized Laplacian with arbitrary real powers on Cayley trees and analyzes periodic solutions of the associated fractional Schr"odinger equations.

Findings

Criteria for eigenvalues allowing periodic solutions for finite index subgroups

Description of a broad class of solutions for infinite index normal subgroups

Detailed analysis of fractional Schr"odinger operators with 0<α<2

Abstract

In this paper we define a discrete generalized Laplacian with arbitrary real power on a Cayley tree. This Laplacian is used to define a discrete generalized Schr\"odinger operator on the tree. The case discrete fractional Schr\"odinger operators with index $0 < \alpha < 2$ is considered in detail, and periodic solutions of the corresponding fractional Schr\"odinger equations are described. This periodicity depends on a subgroup of a group representation of the Cayley tree. For any subgroup of finite index we give a criterion for eigenvalues of the Schr\"odinger operator under which periodic solutions exist. For a normal subgroup of infinite index we describe a wide class of periodic solutions.