Geodesic
Exactly solvable two-dimensional complex model with a real spectrum
Teoretičeskaâ i matematičeskaâ fizika, Tome 148 (2006) no. 1, pp. 102-111 Cet article a éte moissonné depuis la source Math-Net.Ru

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Using supersymmetric intertwining relations of the second order in derivatives, we construct a two-dimensional quantum model with a complex potential for which all energy levels and the corresponding wave functions are obtained analytically. This model does not admit separation of variables and can be considered a complexified version of the generalized two-dimensional Morse model with an additional $\sinh^{-2}$ term. We prove that the energy spectrum of the model is purely real. To our knowledge, this is a rather rare example of a nontrivial exactly solvable model in two dimensions. We explicitly find the symmetry operator, describe the biorthogonal basis, and demonstrate the pseudo-Hermiticity of the Hamiltonian of the model. The obtained wave functions are simultaneously eigenfunctions of the symmetry operator.
Keywords: supersymmetric quantum mechanics, intertwining relations, complex potentials. 
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M. V. Ioffe; F. Cannata; D. N. Nishnianidze. Exactly solvable two-dimensional complex model with a real spectrum. Teoretičeskaâ i matematičeskaâ fizika, Tome 148 (2006) no. 1, pp. 102-111. http://geodesic.mathdoc.fr/item/TMF_2006_148_1_a8/

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