Transparent Scatterers and Transmission Eigenvalues

P. G. Grinevich; R. G. Novikov

doi:10.1007/978-3-031-87213-6_33

Transparent Scatterers and Transmission Eigenvalues

P. G. Grinevich
, R. G. Novikov

Steklov Mathematical Institute of Russian Academy of Sciences
Landau Institute for Theoretical Physics, Russian Academy of Sciences
Department of Mechanics and Mathematics
Moscow State University

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

Abstract

We give a short review of old and recent results on scatterers with transmission eigenvalues of infinite multiplicity, including transparent scatterers. Historically, these studies go back to the publications: Regge (Nuovo Cimento 14, 1959), Newton (J. Math. Phys. 3, 1962) and Sabatier (J. Math. Phys. 7, 1966). Our review is based on the works: Grinevich, Novikov (Commun. Math. Phys. 174, 1995; Eurasian Journal of Mathematical and Computer Applications 9(4), 2021; Russian Math. Surveys, 77(6), 2022). Results of the first of these works include examples of transparent at fixed energy potentials from the Schwartz class in two dimensions. The two others works include the result that, for compactly supported multipoint potentials of Bethe–Peierls–Thomas type in two and three dimensions, any positive energy is a transmission eigenvalue of infinite multiplicity.

Original language	English
Title of host publication	Trends in Mathematics
Publisher	Springer Science and Business Media Deutschland GmbH
Pages	265-274
Number of pages	10
DOIs	https://doi.org/10.1007/978-3-031-87213-6_33
Publication status	Published - 1 Jan 2025

Publication series

Name	Trends in Mathematics
Volume	11
ISSN (Print)	2297-0215
ISSN (Electronic)	2297-024X

Keywords

Multipoint scatterers
Schrödinger’s equation
Spectral problems
Transmisson eigenvalues
Transparent potentials

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10.1007/978-3-031-87213-6_33

Cite this

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AU - Novikov, R. G.

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N2 - We give a short review of old and recent results on scatterers with transmission eigenvalues of infinite multiplicity, including transparent scatterers. Historically, these studies go back to the publications: Regge (Nuovo Cimento 14, 1959), Newton (J. Math. Phys. 3, 1962) and Sabatier (J. Math. Phys. 7, 1966). Our review is based on the works: Grinevich, Novikov (Commun. Math. Phys. 174, 1995; Eurasian Journal of Mathematical and Computer Applications 9(4), 2021; Russian Math. Surveys, 77(6), 2022). Results of the first of these works include examples of transparent at fixed energy potentials from the Schwartz class in two dimensions. The two others works include the result that, for compactly supported multipoint potentials of Bethe–Peierls–Thomas type in two and three dimensions, any positive energy is a transmission eigenvalue of infinite multiplicity.

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KW - Multipoint scatterers

KW - Schrödinger’s equation

KW - Spectral problems

KW - Transmisson eigenvalues

KW - Transparent potentials

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ER -

Transparent Scatterers and Transmission Eigenvalues

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