Coordinate representation for non Hermitian position and momentum operators

Abstract

In this paper we undertake an analysis of the eigenstates of two non self-adjoint operators $\hat q$ and $\hat p$ similar, in a suitable sense, to the self-adjoint position and momentum operators $\hat q_0$ and $\hat p_0$ usually adopted in ordinary quantum mechanics. In particular we discuss conditions for these eigenstates to be {\em biorthogonal distributions}, and we discuss few of their properties. We illustrate our results with two examples, one in which the similarity map between the self-adjoint and the non self-adjoint is bounded, with bounded inverse, and the other in which this is not true. We also briefly propose an alternative strategy to deal with $\hat q$ and $\hat p$, based on the so-called {\em quasi *-algebras}.