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

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.