Exactly Solvable Two-Dimensional Complex Model with Real Spectrum

Cannata, F.; Ioffe, M. V.; Nishnianidze, D. N.

Exactly Solvable Two-Dimensional Complex Model with Real Spectrum

Authors

Cannata, F.

Ioffe, M. V.

Nishnianidze, D. N.

Description

Supersymmetrical intertwining relations of second order in derivatives allow to construct a two-dimensional quantum model with complex potential, for which {\it all} energy levels and bound state wave functions are obtained analytically. This model {\it 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 $\sinh^{-2}$ 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.
13 pages