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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 ([email protected])
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 ([email protected])
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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 

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