Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-19T02:35:36.331Z Has data issue: false hasContentIssue false
  • English
  • Français

Norm resolvent convergence of singularly scaled Schrödinger operators and δ′-potentials

Published online by Cambridge University Press:  17 July 2013

Yu. D. Golovaty  and
R. O. Hryniv
Yu. D. Golovaty
Affiliation:
Department of Mechanics and Mathematics, Ivan Franko National University of Lviv, 1 Universytetska Street, 79000 Lviv, Ukraine (yu_holovaty@franko.lviv.ua)
R. O. Hryniv
Affiliation:
Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova Street, 79060 Lviv, Ukraine and Institute of Mathematics, The University of Rzeszów, 16 Aleja Tadeusza Rejtana, 35-959 Rzeszów, Poland (rhryniv@iapmm.lviv.ua)
Get access Rights & Permissions [Opens in a new window]

Abstract

For a real-valued function V of the Faddeev–Marchenko class, we prove the norm-resolvent convergence, as ε → 0, of a family Sε of one-dimensional Schrödinger operators on the line of the form

Under certain conditions, the functions ε−2V (x/ε) converge in the sense of distributions as ε → 0 to δ′ (x), and then the limit S0 of Sε may be considered as a ‘physically motivated’ interpretation of the one-dimensional Schrödinger operator with potential δ′.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 143 , Issue 4 , August 2013 , pp. 791 - 816
Copyright
Copyright © Royal Society of Edinburgh 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)