Abstract
The eigenvalue problem for a perturbed two-dimensional resonant oscillator is considered. The exciting potential is given by a nonlocal nonlinearity of Hartree type with smooth self-action potential. To each representation of the rotation algebra corresponds the spectral cluster around an energy level of the unperturbed operator. Asymptotic eigenvalues and asymptotic eigenfunctions close to the lower boundary of spectral clusters are obtained. For their calculation, asymptotic formulas for quantum means are used.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. N. Bogolyubov, “On a new form of the adiabatic theory of disturbances in the problem of interaction of particles with a quantum field,” Ukrain. Mat. Zh. 2 (2), 3–24 (1950).
L. P. Pitaevskii, “Bose–Einstein condensation in magnetic traps. Introduction to the theory,” Uspekhi Fiz. Nauk 168 (6), 641–653 (1998). [Physics-Uspekhi 41 (6), 569–580 (1998)].
A. S. Davydov, Solitons inMolecular Systems (Naukova Dumka, Kiev, 1984) [in Russian].
V. P. Maslov, The Complex WKB Method for Nonlinear Equations. I. Linear Theory (Nauka, Moscow, 1977; Birkhäuser Verlag, Basel, 1994).
M. V. Karasev, Quantum Reduction onto Orbits of Symmetry Algebras and the Ehrenfest Problem, Preprint ITF-87-157R (ITF AN UkrSSR, Kiev, 1987) [in Russian].
M. V. Karasev and A. V. Pereskokov, “Asymptotic solutions of Hartree equations concentrated near low-dimensional submanifolds. I: The model with logarithmic singularity,” Izv. Ross. Akad. Nauk Ser. Mat. 65 (5), 33–72 (2001) [Izv. Math. 65 (5), 883–921 (2001)].
M. V. Karasev and A. V. Pereskokov, “Asymptotic solutions of Hartree equations concentrated near low-dimensional submanifolds. II: Localization in planar discs,” Izv. Ross. Akad. Nauk Ser. Mat. 65 (6), 57–98 (2001) [Izv. Math. 65 (6), 1127–1168 (2001)].
V. V. Belov, F. N. Litvinets, and A. Yu. Trifonov, “Semiclassical spectral series of a Hartree-type operator corresponding to a rest point of the classical Hamilton–Ehrenfest system,” Teoret. Mat. Fiz. 150 (1), 26–40 (2007) [Theoret. and Math. Phys. 150 (1), 21–33 (2007)].
V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Short-Wave Diffraction Problems, Vol. 1: The Method of Canonical Problems, With the collaboration of M. M. Popov and I. A. Molotkov (Nauka, Moscow, 1972) [in Russian].
M. V. Karasev and V. P. Maslov, “Asymptotic and Geometric Quantization,” UspekhiMat. Nauk 39 (6 (240)), 115–173 (1984) [Russian Math. Surveys 39 (6), 133–205 (1984)].
M. V. Karasev, “Birkhoff resonances and quantum ray method,” in Days on Diffraction’ 2004, Proceedings of the International Seminar (St. Petersburg, 2004), pp. 114–126.
M. Karasev, “Noncommutative algebras, nanostructures, and quantum dynamics generated by resonances,” in Quanum Algebras and Poisson Geometry in Mathematical Phisics, Amer. Math. Soc. Trans. Ser. 2 (Amer. Math. Soc., Providence, RI, 2005), Vol. 216, pp. 1–17; “Noncommutative algebras, nanostructures, and quantum dynamics generated by resonances. II,” Adv. Stud. Contemp. Math. (Kyungshang) 11 (1), 33–56 (2005); “Noncommutative algebras, nano-structures, and quantum dynamics generated by resonances. III,” Russ. J. Math. Phys. 13 (2), 131–150 (2006).
A. V. Pereskokov, “Asymptotics of the spectrum and quantum averages near the boundaries of spectral clusters for perturbed two-dimensional oscillators,” Mat. Zametki 92 (4), 583–596 (2012) [Math. Notes 92 (3–4), 532–543 (2012)].
A. V. Pereskokov, “Asymptotics of the spectrumand quantum averages of a perturbed resonant oscillator near the boundaries of spectral clusters,” Izv. Ross. Akad. Nauk Ser. Mat. 77 (1), 165–210 (2013) [Izv. Math. 77 (1), 163–210 (2013)].
A. V. Pereskokov, “Semiclassical asymptotic spectrum of a Hartree-type operator near the upper boundary of spectral clusters,” Teoret. Mat. Fiz. 178 (1), 88–106 (2014) [Theoret. and Math. Phys. 178 (1), 76–92 (2014)].
A. V. Pereskokov, “Semiclassical Asymptotics of the spectrum near the upper boundary of spectral clusters for a Hartree-type operator,” Nanostruktury, Mat. Fiz., Modelir. 10 (1), 77–112 (2014).
V. V. Golubev, Lectures in the Analytic Theory of Differential Equations (GITTL, Moscow–Leningrad, 1950) [in Russian].
A. Weinstein, “Asymptotics of the eigenvalues clusters for the Laplasian plus a potential,” Duke Math. J. 44 (4), 883–892 (1977).
J. Schwinger, “On angular momentum,” in Quantum Theory of Angular Momentum (Academic Press, New York, 1965), pp. 229–279.
M. Karasev and E. Novikova, “Non-Lie permutation relations, coherent states and quantum embedding,” in Coherent Transform, Quantization, and Poisson Geometry, Amer. Math. Soc. Trans. Ser. 2 (Amer. Math. Soc., Providence, RI, 1998), Vol. 187, pp. 1–202.
H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 2: Bessel Functions, Parabolic Cylinder Functions, Orthogonal Polynomials (McGraw–Hill, New York–Toronto–London, 1953; Nauka, Moscow, 1974).
M. V. Fedoryuk, Asymptotic Methods for Linear Ordinary Differential Equations (Nauka, Moscow, 1983) [in Russian]; English transl.: Asymptotic Analysis: Linear Ordinary Differential Equations (Springer-Verlag, Berlin, 1993).
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Ed. by M. Abramowitz and I. Stegun (Dover Publications, New York, 1972; Nauka, Moscow, 1979).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A. V. Pereskokov, 2017, published in Matematicheskie Zametki, 2017, Vol. 101, No. 6, pp. 894–910.
Rights and permissions
About this article
Cite this article
Pereskokov, A.V. Semiclassical asymptotics of the spectrum near the lower boundary of spectral clusters for a Hartree-type operator. Math Notes 101, 1009–1022 (2017). https://doi.org/10.1134/S0001434617050285
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434617050285