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^0andp^0usually adopted in ordinary quantum mechanics. In particular, we discuss conditions for these eigenstates to bebiorthogonal 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-calledquasi *-algebras.
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