Smooth approximation of singular perturbations
type: Item
instance of: scholarly article
title: Smooth approximation of singular perturbations
zbMATH Open document ID: 1031.47051
DOI: 10.1016/S0022-247X(03)00163-X
author: Walter B. III Huddell
author: Rhonda J. Hughes
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-07-30T00:00:00Z
review text: One of the main advantages gained by the modern use of point interactions in mathematical physics is the exact solvability of the Schrödinger equation and, simultaneously, the possibility of understanding the underlying formulae simply in the language of the essentially self-adjoint extensions. Unfortunately, the empirically founded interpretation of real systems often prefers the smooth-function form of the interactions. The paper under review shows in which sense the two languages may be used as equivalent. More precisely, in one dimension and for both finite and infinite numbers \(N\) of ``points'', the authors show how the mutual equivalence can be understood in the strong resolvent sense, and also how one can keep the large-\(N\) convergence under strict control. Noticing, in such a context, that the recent development in the field moves towards a weakening of the Hermiticity to mere pseudo-Hermiticity [see, e.g., \textit{S. Albeverio, S. M. Fei} and \textit{P. Kurasov}, Lett. Math. Phys 59, 227-242 (2002; Zbl 1053.81026)], one can only hope that the rigorous studies of the presented type will be also extended in this direction in the nearest future.
reviewed by: Miloslav Znojil
Mathematics Subject Classification ID: 47N50
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 47D08
zbMATH DE Number: 1957231
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: point interactions
zbMATH Keywords: convergence in strong resolvent sense
zbMATH Keywords: approximation by smooth potentials
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1957231