Abstract:
We consider the eigenvalue problem for the three-dimensional Hartree equation in an external field and construct asymptotic (quasi-classical) solutions concentrated near two-dimensional
planar discs. The rate of decrease of these solutions along the normal to the disc is determined by the Bogolyubov polaron, and near the edge of the disc it is defined by the Airy analogue of the polaron. To find the related series of eigenvalues, an analogue of the Bohr–Sommerfeld quantization rule is found from which is derived a simpler algebraic equation determining the main terms in the asymptotics of the eigenvalues.
Citation:
M. V. Karasev, A. V. Pereskokov, “Asymptotic solutions of Hartree equations concentrated near low-dimensional submanifolds. II. Localization in planar discs”, Izv. Math., 65:6 (2001), 1127–1168
This publication is cited in the following 15 articles:
A. V. Pereskokov, “Asymptotic Solutions to the Hartree Equation Near a Sphere. Asymptotics of Self-Consistent Potentials”, J Math Sci, 276:1 (2023), 154
A. V. Pereskokov, “Asymptotics of the spectrum of a Hartree-type operator with a screened Coulomb self-action potential near the upper boundaries of spectral clusters”, Theoret. and Math. Phys., 209:3 (2021), 1782–1797
Shapovalov A.V., Kulagin A.E., Trifonov A.Yu., “The Gross-Pitaevskii Equation With a Nonlocal Interaction in a Semiclassical Approximation on a Curve”, Symmetry-Basel, 12:2 (2020), 201
D. A. Vakhrameeva, A. V. Pereskokov, “Asymptotics of the spectrum of a two-dimensional Hartree-type operator with a Coulomb self-action potential near the lower boundaries of spectral clusters”, Theoret. and Math. Phys., 199:3 (2019), 864–877
A. V. Pereskokov, “Semiclassical Asymptotics of the Spectrum near the Lower Boundary of Spectral Clusters for a Hartree-Type Operator”, Math. Notes, 101:6 (2017), 1009–1022
A. V. Pereskokov, “Semiclassical Asymptotics of Solutions to Hartree Type Equations Concentrated on Segments”, J Math Sci, 226:4 (2017), 462
A. V. Pereskokov, “Semiclassical asymptotic approximation of the two-dimensional Hartree operator spectrum near the upper boundaries of spectral clusters”, Theoret. and Math. Phys., 187:1 (2016), 511–524
A. V. Pereskokov, “Asymptotics of the Hartree operator spectrum near the upper boundaries of spectral clusters: Asymptotic solutions localized near a circle”, Theoret. and Math. Phys., 183:1 (2015), 516–526
A. V. Pereskokov, “Semiclassical asymptotic spectrum of a Hartree-type operator near the upper boundary of spectral clusters”, Theoret. and Math. Phys., 178:1 (2014), 76–92
Lipskaya A.V., Pereskokov A.V., “Asimptoticheskie resheniya odnomernogo uravneniya khartri s negladkim potentsialom vzaimodeistviya. asimptotika kvantovykh srednikh”, Vestnik moskovskogo energeticheskogo instituta, 2012, no. 6, 105–116
Asymptotic solution of the one-dimensional hartree equation with the non-smooth interaction potential. asymtotics of quantum averages
Lipskaya A.V., Pereskokov A.V., “Ob asimptoticheskikh resheniyakh uravneniya tipa khartri s potentsialom vzaimodeistviya yukavy, sosredotochennykh v share”, Vestnik Moskovskogo energeticheskogo instituta, 2011, no. 6, 30–38
On asymptotic solutions concentrated in a ball of the hartree-type equation with the yukawa interaction potential
V. V. Belov, F. N. Litvinets, A. Yu. Trifonov, “Semiclassical spectral series of a Hartree-type operator corresponding
to a rest point of the classical Hamilton–Ehrenfest system”, Theoret. and Math. Phys., 150:1 (2007), 21–33
M. V. Karasev, A. V. Pereskokov, “Asymptotic solutions for Hartree equations and logarithmic obstructions for higher corrections of semiclassical approximation”, Proc. Steklov Inst. Math. (Suppl.), 2003no. , suppl. 1, S123–S128
V. V. Belov, A. Yu. Trifonov, A. V. Shapovalov, “Semiclassical Trajectory-Coherent Approximations of Hartree-Type Equations”, Theoret. and Math. Phys., 130:3 (2002), 391–418
A. V. Pereskokov, “Asymptotic Solutions of Two-Dimensional Hartree-Type Equations Localized in the Neighborhood of Line Segments”, Theoret. and Math. Phys., 131:3 (2002), 775–790
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