Exactly Solvable Non-Separable and Non-Diagonalizable 2-Dim Model with Quadratic Complex Interaction

Abstract

We study a quantum model with non-isotropic two-dimensional oscillator potential but with additional quadratic interaction $x_1{x}_2$ with imaginary coupling constant. It is shown, that for a specific connection between coupling constant and oscillator frequences, the model {\it is not} amenable to a conventional separation of variables. The property of shape invariance allows to find analytically all eigenfunctions and the spectrum is found to be equidistant. It is shown that the Hamiltonian is non-diagonalizable, and the resolution of the identity must include also the corresponding associated functions. These functions are constructed explicitly, and their properties are investigated. The problem of $R-$separation of variables in two-dimensional systems is discussed.

20 pages; minor corrections were made; new Appendix was added

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• Non-hermitean hamiltonians with unitary and antiunitary symmetry
• Is space-time symmetry a suitable generalization of parity-time symmetry?
• Non-Hermitian oscillators with $T_d$ symmetry
• Equidistance of the Complex 2-Dim Anharmonic Oscillator Spectrum: Exact Solution
• Three-Dimensional Shape Invariant Non-Separable Model With Equidistant Spectrum
• Algebraic analysis of non-Hermitian quadratic Hamiltonians
• Dynamical symmetry algebra of two superintegrable two-dimensional systems
• Markov generators as non-hermitian supersymmetric quantum Hamiltonians: spectral properties via bi-orthogonal basis and Singular Value Decompositions