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# Loughborough University Institutional Repository

 Please use this identifier to cite or link to this item: https://dspace.lboro.ac.uk/2134/21400

 Title: Periodic solutions of generalized Schrödinger equations on Cayley Trees Authors: Hiroshima, FumioLorinczi, JozsefRozikov, Utkir Keywords: Cayley treeFractional LaplacianNon-local lattice Schrödinger equationPeriodic solutionsGroup representations Cayley trees Issue Date: 2015 Publisher: Serial Publications Citation: 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. 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. Description: 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 Version: Accepted for publication URI: https://dspace.lboro.ac.uk/2134/21400 Publisher Link: https://www.math.lsu.edu/cosa/index.htm Appears in Collections: Published Articles (Maths)

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