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arXiv:1109.0605 · Journal of Physics A Mathematical and Theoretical

Nonlinear pseudo-bosons versus hidden Hermiticity

Fabio Bagarello, Miloslav Znojil

Published Sep 23, 2011• 21 citations
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Abstract

The increasingly popular concept of a ‘hidden’ Hermiticity of operators is compared with the recently introduced notion of nonlinear pseudo-bosons . The formal equivalence between these two notions is deduced under very general assumptions. Examples of their applicability in quantum mechanics are discussed.
DOI
10.1088/1751-8113/44/41/415305

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We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudo-supersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler-DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.
In the past years several extensions of the canonical commutation relations have been proposed by different people in different contexts and some interesting physics and mathematics have been deduced. Here, we review some recent results on the so-called pseudo-bosons. They arise from a special deformation of the canonical commutation relation [a,a †]= ll, which is replaced by [a,b]=ll, with b not necessarily equal to a †. We start discussing some of their mathematical properties and then we discuss several examples.