Coordinate representation for non Hermitian position and momentum operators

TL;DR

This paper analyzes the eigenstates of non self-adjoint position and momentum operators, exploring their properties, biorthogonality, and examples, with implications for quantum mechanics and alternative algebraic frameworks.

Contribution

It introduces a detailed analysis of eigenstates of non-Hermitian operators similar to standard quantum operators, including conditions for biorthogonality and alternative algebraic approaches.

Findings

Eigenstates can be biorthogonal distributions under certain conditions.

Examples show bounded and unbounded similarity maps affect eigenstate properties.

Proposes an alternative approach using quasi *-algebras.

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}.