Exact analytical solutions of the Schrödinger equation for a two dimensional purely sextic double-well potential

Abstract

Exact analytical solutions of the Schrödinger equation for a two-dimensional purely sextic double-well potential are proved to exist for a denumerably infinite set of the geometry parameter of the well. First, the geometry values which allow exact solutions are determined. Then, explicit wave functions and corresponding energies are calculated for the allowed geometry values. Concrete exact solutions are given for the principal quantum number n up to 10. Moreover, some interesting rules for the obtained exact analytical solutions are also given; particularly, the number of negative energy levels for a given geometry parameter is obtained. For analyzing the obtained exact solutions and for their classification by quantum number, we also use numerical calculations by the Feranchuk-Komarov operator method.