Coordinate representation for non-Hermitian position and momentum operators. (13th September 2017)

• Record Type: Journal Article
• Authors: Bagarello, F.; Gargano, F.; Spagnolo, S.; Triolo, S.
• Is Part Of: Proceedings. Volume 473:Number 2205(2017)
• Publication Date: 2017-09-13
• Subjects: position operators -- generalized eigenvectors -- quasi*-algebras; Physical sciences -- Periodicals; Engineering -- Periodicals; Mathematics -- Periodicals; 500
• DOI: 10.1098/rspa.2017.0434
• Languages: English
• ISSNs: 1364-5021

Abstract

Abstract : In this paper, we undertake an analysis of the eigenstates of two non-self-adjoint operatorsq ^ andp ^ similar, in a suitable sense, to the self-adjoint position and momentum operatorsq ^ 0 andp ^ 0 usually adopted in ordinary quantum mechanics. In particular, we discuss conditions for these eigenstates to be biorthogonal distributions, and we discuss a 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 withq ^ andp ^, based on the so-called quasi *-algebras .