Exactly Solvable Two-dimensional Complex Model with Real Spectrum

Supersymmetrical intertwining relations of second order in derivatives allow to construct a two-dimensional quantum model with complex potential, for which all energy levels and bound state wave functions are obtained analytically. This model is not amenable to separation of variables, and it can be considered as a specific complexified version of generalized two-dimensional Morse model with additional sinh term. The energy spectrum of the model is proved to be purely real. To our knowledge, this is a rather rare example of a nontrivial exactly solvable model in two dimensions. The symmetry operator is found, the biorthogonal basis is described, and the pseudo-Hermiticity of the model is demonstrated. The obtained wave functions are found to be common eigenfunctions both of the Hamiltonian and of the symmetry operator. This paper is dedicated to the eightieth birthday of Yuri Victorovich Novozhilov. PACS numbers: 03.65.-w, 03.65.Fd, 11.30.Pb