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Semiclassical asymptotic approximation of the two-dimensional Hartree operator spectrum near the upper boundaries of spectral clusters
Theoretical and Mathematical Physics. 2016. Vol. 187. No. 1. P. 511-524.
We consider an eigenvalue problem for the two-dimensional Hartree operator with a small parameter at the nonlinearity. We obtain the asymptotic eigenvalues and the asymptotic eigenfunctions near the upper boundaries of the spectral clusters formed near the energy levels of the unperturbed operator and construct an asymptotic expansion around the circle where the solution is localized.
Priority areas:
mathematics
Language:
English
A. V. Pereskokov, Theoretical and Mathematical Physics 2015 Vol. 183 No. 1 P. 516-526
We consider the eigenvalue problem for the Hartree operator with a small parameter multiplying the
nonlinearity. We obtain asymptotic eigenvalues and asymptotic eigenfunctions near the upper boundaries
of spectral clusters formed near the energy levels of the unperturbed operator. Near the circle where
the solution is localized, the leading term of the expansion is a solution of the ...
Added: March 6, 2017
A. V. Pereskokov, Theoretical and Mathematical Physics 2020 Vol. 205 No. 3 P. 1652-1665
We consider the eigenvalue problem for the two-dimensional Hartree operator with a small nonlinearity
coefficient. We find the asymptotic eigenvalues and asymptotic eigenfunctions near a local maximum of
the eigenvalues in spectral clusters formed near the eigenvalues of the unperturbed operator. ...
Added: December 7, 2020
A. V. Pereskokov, Journal of Mathematical Sciences 2017 Vol. 226 No. 4 P. 517-530
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. ...
Added: December 21, 2017
Pereskokov A., Теоретическая и математическая физика 2016 Т. 187 № 1 С. 74-87
We consider an eigenvalue problem for the fwo-dimensional Hartree operator with a small parameter at
the nonlinearity. We obtain the asymptotic eigenvalues and the asymptotic eigenfunctions near the upper
boundaries of the spectral clusters formed near the energy levels of the unperturbed operator and construct
an asymptotic expansion around the circle where the solution is localized. ...
Added: April 18, 2016
Pereskokov A., Theoretical and Mathematical Physics 2021 Vol. 209 No. 3 P. 1782-1797
We consider the eigenvalue problem for a Hartree-type operator with a screened Coulomb self-action
potential and with a small parameter multiplying the nonlinearity. We obtain asymptotic eigenvalues and
asymptotic eigenfunctions near the upper boundaries of spectral clusters that form near the energy levels
of the unperturbed operator. ...
Added: December 6, 2021
A. V. Pereskokov, Mathematical notes 2017 Vol. 101 No. 6 P. 1009-1022
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 ...
Added: December 20, 2017
Alexander Pereskokov, Applicable Analysis 2016 Vol. 95 No. 7 P. 1560-1569
We consider the eigenvalue problem for a perturbed two-dimensional
resonance oscillator. The excitation potential is given by a Hartree-type
nonlinearity with a smooth self-action potential. We use asymptotic
formulas for the quantum averages to obtain asymptotic eigenvalues and
asymptotic eigenfunctions near the lower boundaries of spectral clusters
which are formed near the energy levels of the unperturbed operator. ...
Added: March 4, 2017
Pereskokov A., Теоретическая и математическая физика 2015 Т. 183 № 1 С. 78-89
We consider the eigenvalue problem for the Hartree operator with a small parameter multipliplying the
nonlinearity. We obtain asymptotic eigenvalues and asymptotic eigenfunctions near the upper boundaries
of spectral clusters formed near the energy leves of the unperturbed operator. Near the circle where
the solution is localized, the leading term of the expansion is a solution of the ...
Added: April 25, 2015
A. V. Pereskokov, Journal of Mathematical Sciences 2017 Vol. 226 No. 4 P. 462-516
We study the nonlinear eigenvalue problem for two-dimensional Hartree type equations with selfaction potentials possessing logarithmic singularity and depending on the distance between points. To find a series of asymptotic eigenvalues, we derive a counterpart of the Bohr–Sommerfeld quantization rule. The corresponding asymptotic eigenfunctions are localized near a plane segment. ...
Added: December 20, 2017
Pereskokov A., Journal of Mathematical Sciences 2022 Vol. 264 No. 5 P. 617-632
We consider the spectral problem for a perturbed two-dimensional oscillator. The role of a perturbation is played by an integral Hartree type nonlinearity with a self-action potential depending on the distance between points and possessing a Coulomb singularity. We find asymptotic eigenvalues and eigenfunctions near boundaries of spectral clusters appearing near eigenvalues of the unperturbed ...
Added: October 24, 2022
Pereskokov A., Наноструктуры. Математическая физика и моделирование 2014 Т. 11 № 1 С. 45-66
Рассматривается задача на собственные значения для оператора Хартри с кулоновским взаимодействием,
который содержит малый параметр перед нелинейностью. Найдены асимптотические собственные значения и
асимптотические собственные функции вблизи верхних границ спектральных кластеров. Вблизи окружности,
где сосредоточено решение, главный член разложения является решением задачи о двумерном операторе. ...
Added: November 4, 2014
D. A. Vakhrameeva, A. V. Pereskokov, Theoretical and Mathematical Physics 2019 Vol. 199 No. 3 P. 864-877
We consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an
integral Hartree-type nonlinearity with a Coulomb self-action potential. We obtain asymptotic eigenvalues
and asymptotic eigenfunctions near the lower boundaries of spectral clusters formed in a neighborhood of
the eigenvalues of the unperturbed operator and construct an asymptotic expansion near a circle where
the solution ...
Added: May 28, 2019
Vakhrameeva D. A., Pereskokov A. V., Journal of Mathematical Sciences 2020 Vol. 247 No. 6 P. 820-849
We study the spectral problem for a two-dimensional Hartree type operator with smooth selfaction potential. We find
asymptotic eigenvalues and eigenfunctions and construct an asymptotic expansion for quantum averages near
the lower boundaries of spectral clusters. ...
Added: June 22, 2020
A. V. Pereskokov, Russian Journal of Mathematical Physics 2019 Vol. 26 No. 3 P. 391-400
The problem of the Zeemann–Stark effect for the hydrogen atom in electromagnetic
fields is considered using the irreducible representations of the Karasev–Novikova algebra
with quadratic commutation relations. An asymptotics of the series of eigenvalues and
the asymptotic eigenfunctions are obtained near the upper boundaries of resonance spectral
clusters which are formed near the energy levels of an unperturbed hydrogen ...
Added: November 16, 2019
А.В. Перескоков, В кн. : Современные методы теории краевых задач. : Воронеж : Издательский дом ВГУ, 2020. С. 168-169.
Рассматривается задача на собственные значения для нелинейного оператора типа Хартри. Особенностью задачи является то, что она относится к классу резонансных. В работе найдена серия асимптотических собственных значений вблизи верхних границ спектральных кластеров. ...
Added: February 27, 2021
Budkov Y., М. : ЛЕНАНД, 2020
Within the presented monograph for the first time statistical approaches, based on the self-consistent field theory, were presented for the theoretical description of the thermodynamic properties of the ion-molecular systems (electrolyte solutions, ionic liquids, dielectric polymers and metal-organic frameworks) in the bulk solution and at the interfaces with the account for their molecular structure. In ...
Added: November 18, 2019
Pereskokov A., Липская А. В., Вестник Московского энергетического института 2012 № 6 С. 105-116
Рассмотрена задача на собственные значения для одномерного уравнения Хартри, содержащего интегральную нелинейность с негладким потенциалом взаимодействия. Найдены асимптотические собственные значения и собственные функции, локализованные вблизи точки. Изучена задача вычисления средних значений дифференциальных операторов на решениях. ...
Added: December 31, 2012
Pereskokov A., Наноструктуры. Математическая физика и моделирование 2014 Т. 10 № 1 С. 77-112
Рассматривается задача на собственные значения для возмущенного двумерного резонансного осциллятора. Возбуждающий потенциал задается нелокальной нелинейностью типа Хартри с гладким потенциалом самодействия. Каждому представлению алгебры вращений соответствует спектральный кластер вокруг уровня энергии невозмущенного оператора. Найдены асимптотические собственные значения и асимптотические собственные функции вблизи верхних границ спектральных кластеров. Для их вычисления использованы асимптотические формулы для квантовых средних. ...
Added: November 16, 2013
Migaeva A. S., Pereskokov A., Journal of Mathematical Sciences 2020 Vol. 251 No. 6 P. 850-875
We study the Zeeman-Stark effect in the hydrogen atom located in an electromagnetic field by using irreducible
representations of an algebra with the Karasev-Novikova quadratic commutation relations. The representations
are associated with resonance spectral clusters near the energy level of the unperturbed hydrogen atom. We find
asymptotics for a series of eigenvalues and corresponding asymptotic eigenfunctions near the ...
Added: December 7, 2020
Pereskokov A., Вестник Московского энергетического института 2013 № 6 С. 180-190
Рассматривается задача на собственные значения для возмущенного двумерного резонансного осциллятора. Возбуждающий потенциал задается интегральной нелинейностью типа Хартри с гладким потенциалом самодействия. Найдены асимптотические собственные значения и асимптотические собственные функции вблизи верхних границ спектральных кластеров, которые образуются вокруг уровней энергии невозмущенного оператора. Для их вычисления использованы асимптотические формулы для квантовых средних. ...
Added: November 15, 2013
Pereskokov A., Теоретическая и математическая физика 2014 Т. 178 № 1 С. 88-106
Рассматривается задача на собственные значения для возмущенного двумерного осциллятора в случае резонанса частот. Возбуждающий потенциал задается интегральной нелинейностью типа Хартри с гладким потенциалом самодействия. Найдены асимптотические собственные значения и асимптотические собственные функции вблизи верхних границ спектральных кластеров, которые образуются вокруг уровней энергии невозмущенного оператора. Для их вычисления использованы асимптотические формулы квантовых средних. ...
Added: November 16, 2013
Akhmedov E., Physical Review D - Particles, Fields, Gravitation and Cosmology 2013 Vol. 87 No. 4 P. 044049
Following Krotov and Polyakov [ Nucl. Phys. B849 410 (2011)], we show that in global de Sitter space its isometry is broken by the loop IR divergences for any invariant vacuum state of the massive scalars. We derive a kinetic equation in global de Sitter space that follows from the Dyson-Schwinger equation of the Schwinger-Keldysh ...
Added: February 27, 2013
BOSSY M., Jabir J. M., Electronic Communications in Probability 2018 Vol. 23 P. 1-14
In this paper, we prove a particle approximation, in the sense of the propagation of chaos, of a Lagrangian stochastic model submitted to specular boundary condition and satisfying the mean no-permeability condition. ...
Added: June 7, 2018
Sergeev A., Васильев А. Ю., Russian Mathematical Surveys 2013 Vol. 68 No. 3 P. 435-502
Teichmüller theory is a ramified and rapidly developing area of mathematics which has multiple connections with other directions in the mathematical sciences and with their applications, first and foremost in mathematical physics. In this survey the main lines of development of this theory and its applications to string theory are presented in a historical context.
Bibliography: 128 titles. ...
Added: April 9, 2015