A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. It turns out that such operators generate a canonical lattice of Hilbert spaces, that is, the simplest case of a partial inner product space (PIP-space). We introduce several generalizations of the notion of similarity between operators, in particular, the notion of quasi-similarity, and we explore to what extend they preserve spectral properties. Then we apply some of the previous results to operators on a particular PIP-space, namely, a scale of Hilbert spaces generated by a metric operator. Finally, motivated by the recent developments of pseudo-Hermitian quantum mechanics, we reformulate the notion of pseudo-Hermitian operators in the preceding formalism.

http://arxiv.org/abs/1409.3497
Mathematical Physics (math-ph)

We show how to construct, out of a certain basis invariant under the action of one or more unitary operators, a second biorthogonal set with similar properties. In particular, we discuss conditions for this new set to be also a basis of the Hilbert space, and we apply the procedure to coherent states. We conclude the paper considering a simple application of our construction to pseudo-hermitian quantum mechanics.

http://arxiv.org/abs/1402.0425
Mathematical Physics (math-ph)

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 show how some recent models of PT-quantum mechanics perfectly fit into the settings of \(\cal D\) pseudo-bosons, as introduced by one of us. Among the others, we also consider a model of non-commutative quantum mechanics, and we show that this model too can be described in terms of \(\cal D\) pseudo-bosons.

http://arxiv.org/abs/1310.0359
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. Following our previous work, we introduce several generalizations of the notion of similarity between operators. Then we explore systematically the various types of quasi-Hermitian operators, bounded or not. Finally we discuss their application in the so-called pseudo-Hermitian quantum mechanics.

http://arxiv.org/abs/1307.5644
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)