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.

http://arxiv.org/abs/1310.4775

Quantum Physics (quant-ph); Mathematical Physics (math-ph)

We have recently proposed a strategy to produce, starting from a given hamiltonian \(h_1\) and a certain operator \(x\) for which \([h_1,xx^\dagger]=0\) and \(x^\dagger x\) is invertible, a second hamiltonian \(h_2\) with the same eigenvalues as \(h_1\) and whose eigenvectors are related to those of \(h_1\) by \(x^\dagger\). Here we extend this procedure to build up a second hamiltonian, whose eigenvalues are different from those of \(h_1\), and whose eigenvectors are still related as before. This new procedure is also extended to crypto-hermitian hamiltonians.

http://arxiv.org/abs/1110.4828
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

The increasingly popular concept of a hidden Hermiticity of operators (i.e., of their Hermiticity with respect to an {\it ad hoc} inner product in Hilbert space) is compared with the recently introduced notion of {\em non-linear pseudo-bosons}. The formal equivalence between these two notions is deduced under very general assumptions. Examples of their applicability in quantum mechanics are discussed.

http://arxiv.org/abs/1109.0605
Mathematical Physics (math-ph); Functional Analysis (math.FA); Quantum Physics (quant-ph)