Analytic wave functions and the corresponding energies for a class of the $$ \mathcal{P}\mathcal{T} $$-symmetric two-dimensional quartic potentials are found. The general form of the solutions is discussed.
This paper presents a direct algebraic method of searching for analytic solutions of the two-dimensional time-independent Schrödinger equation that is impossible to separate into two one-dimensional ones. As examples, two-dimensional polynomial and Morse-like potentials are discussed. Analytic formulas for the ground state wave functions and the corresponding energies are presented. These results cannot be obtained by another known method.
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