We consider the eigenvalue problem for a two-dimensional perturbed resonance oscillator. The role of perturbation is played by an integral Hartree type nonlinearity, where the selfaction potential depends on the distance between points and has logarithmic singularity. We obtain asymptotic eigenvalues near the upper boundaries of spectral clusters appeared near eigenvalues of the unperturbed operator.
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V. M. Babich and V. S. Buldyrev, Short-Wavelength Diffraction Theory: Asymptotic Methods, Springer, Berlin (1972).
V. P. Maslov, The Complex WKB Method for Nonlinear Equations, Birkhäuser, Basel (1994).
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, “Asymptotics of the Hartree operator spectrum near the upper boundaries of spectral clusters: Asymptotic solutions localized near a circle,” Theor. Math. Phys. 183, No. 1, 516–526 (2015).
V. P. Maslov, Operator Methods [in Russian], Nauka, M. (1973).
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Translated from Problemy Matematicheskogo Analiza 89, July 2017, pp. 163-174.
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Pereskokov, A.V. 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, 517–530 (2017). https://doi.org/10.1007/s10958-017-3545-7
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DOI: https://doi.org/10.1007/s10958-017-3545-7