Metric operators, generalized hermiticity and lattices of Hilbert spaces

Abstract

Pseudo-Hermitian quantum mechanics (QM) is a recent, unconventional, approach to QM, based on the use of non-self-adjoint Hamiltonians, whose self-adjointness can be restored by changing the ambient Hilbert space, via a so-called metric operator. The PT-symmetric Hamiltonians are usually pseudo-Hermitian operators, a term introduced a long time ago by Dieudonné for characterizing those bounded operators A that satisfy a relation of the form GA = A G, where G is a metric operator, that is, a strictly positive self-adjoint operator. This chapter explores further the structure of unbounded metric operators, in particular, their incidence on similarity. It examines the notion of similarity between operators induced by a bounded metric operator with bounded inverse. The goal here is to study which spectral properties are preserved under such a quasi-similarity relation. The chapter applies some of the previous results to operators on the scale of Hilbert spaces generated by the metric operator.