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arXiv:1607.00350 · Banach Journal of Mathematical Analysis

Non-self-adjoint Schrödinger operators with nonlocal one-point interactions

Sergiusz Kużel, Miloslav Znojil

Published Sep 11, 2017• 10 citations
arXivHTMLPDF
DOI
10.1215/17358787-2017-0032

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Recently, we constructed an energy-dependent point interaction (EDPI) in its most general form in one-dimensional quantum mechanics. In this paper, we show that stationary solutions of the Schrodinger equation with the EDPI form a complete set. Then any nonstationary solution of the time-dependent Schrodinger equation can be expressed as a linear combination of stationary solutions. This, however, does not necessarily mean that the EDPI is self-adjoint and the time-development of the nonstationary state is unitary. The EDPI is self-adjoint provided that the stationary solutions are all orthogonal to one another. We illustrate situations in which this orthogonality condition is not satisfied.PACS Nos.: 03.65.–w, 03.65.Nk, 03.65.Ge