A non self-adjoint model on a two dimensional noncommutative space with unbound metric

Abstract

We demonstrate that a non self-adjoint Hamiltonian of harmonic oscillator type defined on a two-dimensional noncommutative space can be diagonalized exactly by making use of pseudo-bosonic operators. The model admits an antilinear symmetry and is of the type studied in the context of PT-symmetric quantum mechanics. Its eigenvalues are computed to be real for the entire range of the coupling constants and the biorthogonal sets of eigenstates for the Hamiltonian and its adjoint are explicitly constructed. We show that despite the fact that these sets are complete and biorthogonal, they involve an unbounded metric operator and therefore do not constitute (Riesz) bases for the Hilbert space $\Lc^2(\Bbb R^2)$, but instead only D-quasi bases. As recently proved by one of us (FB), this is sufficient to deduce several interesting consequences.

11 pages, in press in Physical Review A

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• Classical and quantum dynamics in the (non-Hermitian) Swanson oscillator
• A no-go result for the quantum damped harmonic oscillator
• Some remarks on quasi-Hermitian operators
• Seiberg-Witten map and quantum phase effects for neutral Dirac particle on noncommutatiave plane
• Intertwining operators for non self-adjoint Hamiltonians and bicoherent states
• $\mathcal{D}$-deformed harmonic oscillators
• $\mathcal{D}$-Pseudo-Bosons, Complex Hermite Polynomials, and Integral Quantization
• Entropy and Information of a harmonic oscillator in a time-varying electric field in 2D and 3D noncommutative spaces
• Tridiagonality, supersymmetry and non self-adjoint Hamiltonians
• Quasi-Hermitian formulation of quantum mechanics using two conjugate Schrödinger equations
• Quantum, noncommutative and MOND corrections to the entropic law of gravitation
• Generalized Bogoliubov transformations versus $\mathcal{D}$% -pseudo-bosons
• Features, paradoxes and amendments of perturbative non-Hermitian quantum mechanics