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A characterization of singular Schrödinger operators on the half-line

Part of: Asymptotic theory Real functions: Miscellaneous topics Ordinary differential operators Other (nonclassical) types of operator theory Elliptic equations and systems Nonstandard models

Published online by Cambridge University Press:  07 December 2020

Raffaele Scandone ,
Lorenzo Luperi Baglini  and
Kyrylo Simonov
Raffaele Scandone*
Affiliation:
Gran Sasso Science Institute, Viale F. Crispi 7, 67100 L'Aquila, Italy
Lorenzo Luperi Baglini
Affiliation:
Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133Milano, Italy e-mail: lorenzo.luperi@unimi.it
Kyrylo Simonov
Affiliation:
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090Vienna, Austria e-mail: kyrylo.simonov@univie.ac.at
*
e-mail: raffaele.scandone@gssi.it
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Abstract

We study a class of delta-like perturbations of the Laplacian on the half-line, characterized by Robin boundary conditions at the origin. Using the formalism of nonstandard analysis, we derive a simple connection with a suitable family of Schrödinger operators with potentials of very large (infinite) magnitude and very short (infinitesimal) range. As a consequence, we also derive a similar result for point interactions in the Euclidean space $\mathbb {R}^3$ , in the case of radial potentials. Moreover, we discuss explicitly our results in the case of potentials that are linear in a neighborhood of the origin.

Keywords

Schrödinger operatorssingular perturbationspoint interactionsnonstandard analysis

MSC classification

Primary: 34L40: Particular operators (Dirac, one-dimensional Schrödinger, etc.) 34E15: Singular perturbations, general theory 35J10: Schrödinger operator 03H05: Nonstandard models in mathematics 26E35: Nonstandard analysis
Secondary: 47S20: Nonstandard operator theory
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 4 , December 2021 , pp. 923 - 941
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The research of L.L.B. was supported by grant P 30821-N35 of the Austrian Science Fund FWF, and the research of K.S was supported by grant P 30821-N35 of the Austrian Science Fund FWF.

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