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# Periodic solutions of generalized Schrödinger equations on Cayley Trees

journal contribution
posted on 27.05.2016, 14:42 by Fumio Hiroshima, Jozsef Lorinczi, Utkir Rozikov
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.

• Science

## Department

• Mathematical Sciences

## Published in

published in Communications on Stochastic Analysis 9 (2), 283-296, 2015

## 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.

## Publisher

Serial Publications

## Version

AM (Accepted Manuscript)

## Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

2015

## Notes

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

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