TY - CHAP
T1 - Transparent Scatterers and Transmission Eigenvalues
AU - Grinevich, P. G.
AU - Novikov, R. G.
N1 - Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.
PY - 2025/1/1
Y1 - 2025/1/1
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.
AB - 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.
KW - Multipoint scatterers
KW - Schrödinger’s equation
KW - Spectral problems
KW - Transmisson eigenvalues
KW - Transparent potentials
UR - https://www.scopus.com/pages/publications/105011713267
U2 - 10.1007/978-3-031-87213-6_33
DO - 10.1007/978-3-031-87213-6_33
M3 - Chapter
AN - SCOPUS:105011713267
T3 - Trends in Mathematics
SP - 265
EP - 274
BT - Trends in Mathematics
PB - Springer Science and Business Media Deutschland GmbH
ER -