We study the eigenvalue problem for a perturbed two-dimensional oscillator. The role of a perturbation is played by a Hartree type integral nonlinearity, where the self-consistent potential contains the Macdonald function and depends on the distance between points. We find asymptotic eigenvalues and asymptotic eigenfunctions near the upper boundaries of spectral clusters. We construct an asymptotic expansion near a circle, where the solution is localized.
Similar content being viewed by others
References
V. P. Maslov, The Complex WKB Method for Nonlinear Equations, Birkhäuser, Basel (1994).
I. V. Simenog, “Asymptotics solution of stationary nonlinear Hartree equation,” Theor. Math. Phys. 30, 263–268 (1977).
M. V. Karasev and V. P. Maslov, “Algebras with general commutation relations and their applications. II. Operator unitary-nonlinear equations,” J. Math. Sci. 15, No. 3, 273–368 (1981).
A. V. Pereskokov, “Semiclassical asymptotic approximation of the two-dimensional Hartree operator spectrum near the upper boundaries of spectral clusters,” Theor. Math. Phys. 187, No. 1, 511–524 (2016).
A. V. Pereskokov, “Semiclassical asymptotic spectrum of the two-dimensional Hartree operator near a local maximum of the eigenvalues in a spectral cluster,” Theor. Math. Phys. 205, No. 3, 1652–1665 (2020).
A. V. Pereskokov, “Asymptotics of the spectrum of a two-dimensional Hartree type operator near upper boundaries of spectral clusters. Asymptotic solutions located near a circle,” J. Math. Sci. 226, No. 4, 517–530 (2017).
P. K. Suetin, Classical Orthogonal Polynomials [in Russian], Fizmatlit, Moscow (2007).
V. P. Maslov, Operator Methods [in Russian], Nauka, Moscow (1973).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Problemy Matematicheskogo Analiza 126, 2024, pp. 51-64.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Pereskokov, A.V. Asymptotics of the Spectrum of a Hartree Type Operator with Self-Consistent Potential Including the Macdonald Function. J Math Sci 279, 508–524 (2024). https://doi.org/10.1007/s10958-024-07029-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-024-07029-9