Periodic Solutions of Generalized Schrödinger 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 Schrödinger operator on the tree. The case discrete fractional Schrödinger operators with index $0<\alpha <2$ is considered in detail, and periodic solutions of the corresponding fractional Schrödinger 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 Schrödinger operator under which periodic solutions exist. For a normal subgroup of infinite index we describe a wide class of periodic solutions.

14 pages, 3 figures. To appear in Commun. Stoch. Anal

References in corpus (2)

• Lieb-Thirring Bound for Schrödinger Operators with Bernstein Functions of the Laplacian
• Pointwise eigenfunction estimates and intrinsic ultracontractivity-type properties of Feynman-Kac semigroups for a class of Lévy processes