Transparent scatterers and transmission eigenvalues
P.G. Grinevich (1, 2, 3), R.G. Novikov (4) ((1) Steklov, Mathematical Institute, RAS, Moscow, Russia, (2) L.D. Landau Institute for, Theoretical Physics, RAS, Moscow Region, Russia, (3) Lomonosov Moscow State, University, Department of Mechanics, Mathematics, Moscow, Russia, (4)
TL;DR
This paper reviews the mathematical theory of transparent scatterers and transmission eigenvalues, highlighting classical and recent results on potentials with infinite multiplicity eigenvalues, including explicit examples and general conditions.
Contribution
It synthesizes historical and recent research on transmission eigenvalues, emphasizing the existence of transparent scatterers with infinite multiplicity eigenvalues in various dimensions.
Findings
Examples of transparent potentials in two dimensions.
Positive energies as eigenvalues of infinite multiplicity.
Connection between multipoint potentials and transmission eigenvalues.
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.
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