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Exactly Solvable Two-Dimensional Complex Model with Real Spectrum

F. Cannata·M. V. Ioffe·D. N. Nishnianidze
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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.

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F. Cannata, M. V. Ioffe, D. N. Nishnianidze. 2005-12-09. Exactly Solvable Two-Dimensional Complex Model with Real Spectrum. https://arxiv.org/abs/hep-th/0512110

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