Exactly Solvable Two-Dimensional Complex Model with Real Spectrum

TL;DR

This paper introduces a two-dimensional complex quantum model with a real energy spectrum, analytically solvable despite not allowing variable separation, and demonstrates its pseudo-Hermiticity and symmetry properties.

Contribution

It presents a novel exactly solvable 2D complex quantum model with real spectrum, constructed via supersymmetrical intertwining, not separable, and with explicit wave functions and symmetry operator.

Findings

All energy levels and bound states are analytically obtained.

The model's spectrum is proven to be purely real.

The model exhibits pseudo-Hermiticity and a symmetry operator.

Abstract

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.