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Download filePeriodic solutions of generalized Schrödinger equations on Cayley Trees
journal contribution
posted on 2016-05-27, 14:42 authored by Fumio Hiroshima, Jozsef Lorinczi, Utkir RozikovIn 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.
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