Abstract
The stationary Schrödinger equation is studied in a domain bounded by two confocal ellipses and in its coverings. The order of dependence of the Laplace operator eigenvalues on sufficiently small distance between the foci is obtained. Coefficients of the power series expansion of said eigenvalues are calculated up to and including the square of half the focal distance.
This is a preview of subscription content, log in via an institution to check access.
Notes
We can consider the Mathieu equation as a eigenvalue problem for the operator \(D(y)=\dfrac{d^{2}y}{dx^{2}}-2q\cos(2x)y\) (or operator \(D(y)=\dfrac{d^{2}y}{dx^{2}}-2q\cosh(2x)y\)). Therefore, in the literature \(a\) is often called the eigenvalues.
REFERENCES
V. V. Fokicheva, ‘‘Description of singularities for billiard systems bounded by confocal ellipses or hyperbolas,’’ 68, 148–157 (2014). https://doi.org/10.3103/S0027132214040020
V. V. Fokicheva, ‘‘A topological classification of billiards in locally planar domains bounded by arcs of confocal quadrics,’’ Sb. Math. 206, 1463–1507 (2015). https://doi.org/10.1070/SM2015v206n10ABEH004502
V. V. Vedyushkina (Fokicheva) and A. T. Fomenko, ‘‘Integrable geodesic flows on orientable two-dimensional surfaces and topological billiards,’’ Izv. Math. 83, 1137–1173 (2019). doi 10.1070/IM8863
R. F. A. Clebsch, Theorie der Elasticität fester Körper (B.G. Teubner, Leipzig, 1921).
J. W. S. B. Lord Rayleigh, The Theory of Sounds (Dover, New York, 1956).
J. R. Kuttler and V. G. Sigillito, ‘‘Eigenvalues of the Laplacian in two dimensions,’’ SIAM Rev. 26, 163–193 (1984). https://doi.org/10.1137/1026033
M. Abramowitz, I. A. Stegun, and D. Miller, in Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (U.S. Dept. of Commerce National Bureau of Standards, 1972), pp. 721–750.
J. McMahon, ‘‘On the roots of the Bessel and certain related functions,’’ Ann. Math. 9, 230–230 (1894). https://doi.org/10.2307/1967501
N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford Univ. Press, Oxford, 1951).
‘‘Mathieu functions and Hill’s equation,’’ in NIST Digital Library of Mathematical Functions, Ed. by F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, et al. (NIST, 2023). https://dlmf.nist.gov/28.
J. Meixner and F. W. Schafke, ‘‘Mathieusche Funktionen,’’ in Mathieusche Funktionen und Spharoidfunktionen, Die Grundlehren der Mathematischen Wissenschaften, Vol. 71 (Springer, Berlin, 1954), pp. 98–221. https://doi.org/10.1007/978-3-662-00941-3_3
ACKNOWLEDGMENTS
The author thanks Member of the Russian Academy of Sciences A.T. Fomenko for his permanent attention to the work. The author thank the anonymous reviewer for careful reading of the manuscript and fruitful remarks that allow improving the exposition.
Funding
The work is carried out at the Lomonosov Moscow State University and is supported by the Russian Science Foundation, project no. 22-71-10106.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author of this work declares that he has no conflicts of interest.
Additional information
Translated by E. Oborin
Publisher’s Note. Allerton Press remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Nikulin, M.A. Spectrum of Schrödinger Operator in Covering of Elliptic Ring. Moscow Univ. Math. Bull. 78, 230–243 (2023). https://doi.org/10.3103/S0027132223050042
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0027132223050042