Cayley tree Fractional Laplacian Non-local lattice Schrödinger equation Periodic solutions Group representations Cayley trees
HIROSHIMA, F., LORINCZI, J. and ROZIKOV, U., Periodic Solutions of Generalized Schrödinger Equations on Cayley Trees. Communications on Stochastic Analysis 9(2), pp. 283-296.
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.
This paper was accepted for publication in the journal Communications on Stochastic Analysis and the definitive published version is available at https://www.math.lsu.edu/cosa/index.htm