Asymptotics of the Spectrum of a Hartree Type Operator with Self-Consistent Potential Including the Macdonald Function

?

Asymptotics of the Spectrum of a Hartree Type Operator with Self-Consistent Potential Including the Macdonald Function

Journal of Mathematical Sciences. 2024. Vol. 279. No. 4. P. 508–524.

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.

Research target: Mathematics

Language: English

Full text

DOI

Space-Time Fluctuations in a Quasi-static Limit

Bernardin C., Gonçalves P., Olla S., Mathematical Physics Analysis and Geometry 2024 Vol. 27 No. 7

We consider the macroscopic limit for the space-time density fluctuations in the open symmetric simple exclusion in the quasi-static scaling limit. We prove that the distribution of these fluctuations converge to a gaussian space-time field that is delta correlated in time but with long-range correlations in space. ...

Added: October 6, 2026

To spike or not to spike: the whims of the Wonham filter in the strong noise regime

Bernardin C., Chhaibi R., Najnudel J. et al., Probability Theory and Related Fields 2026 Vol. 195 P. 1823–1875

We study the celebrated Shiryaev-Wonham filter (Wonham, W.M., in J. Soc. Ind. Appl. Math. 347–369, 1964) in its historical setup, where the hidden Markov jump process has two states. We are interested in the weak noise regime for the observation equation. Interestingly, this becomes a strong noise regime for the filtering equations. Earlier results of ...

Added: October 5, 2026

Цепная дробь Аски–Вильсона при qN=1

Ismailov A., Spiridonov V., Успехи математических наук 2026 Т. 81 № 5 С. 183–184

Получена новая формула для цепной дроби Аски–Вильсона в форме отношения двух q-гипер-геометрических рядов. ...

Added: October 5, 2026

Explicit Formula for Inverse and Determinant in Geometric Algebras over Odd-dimensional Vector Spaces

Abdulkhaev K., Shirokov D., Advances in Applied Clifford Algebras 2026 Vol. 36 P. 1–21

In this paper, we present explicit formulas for the inverse and determinant in geometric (Cliﬀord) algebras over vector spaces of dimension n = 7. The derivation of these formulas is made possible by generalizing the concept of conjugation to basis conjugation operations. We further develop a general method for constructing such formulas over odd-dimensional spaces ...

Added: October 4, 2026

On Practical Aspects of Constructing Quasi-Cyclic Subfield Subcodes of Dual Elliptic Codes and Their Application in McEliece-type Cryptosystems

Kuninets A., IEEE Transactions on Information Theory 2026 P. 1–1

In this work we study the applicability of Quasi-Cyclic Subfield Subcodes of Dual Elliptic (QC-SSDE) codes for integration into code-based cryptographic schemes. Detailed algorithms are provided for constructing parity-check matrices as well as block-circulant parity-check matrices for this family of codes, accompanied by empirical results that enable the construction of QC-SSDE codes with predetermined dimensions. ...

Added: October 3, 2026

Graphon spin systems as exactly solvable models

Medvedev G., Alexandrov Artem, Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2026 Vol. 114 Article 044102

Graphons are measurable functions used to describe the asymptotic behavior of convergent graph families. Originally motivated by problems in combinatorics and graph theory, graphons have found numerous applications in the modeling and analysis of dynamical processes on networks. In this work, we use graphons to formulate the Ising model on convergent graph sequences, which include ...

Added: October 2, 2026

Планетарное зацепление трилистника

Pochinka O., Baranov D., Nozdrinova E., Теоретическая и математическая физика 2026 Т. 229 № 1 С. 3–14

The Birman–Williams problem on describing the planetary link of a fibered knot K in S^3 has been partially solved. Using Nielsen's theory for the classification of periodic surface homeomorphisms and its close relationship with the theory of gradient-like diffeomorphisms, it is proved that the planetary link of the trefoil (the unique periodic fibered knot of genus ...

Added: October 2, 2026

A quantum–analogue formalism for modeling supraliminal information processing

Lubashevsky I., Lubashevskiy V., Physica D: Nonlinear Phenomena 2026 Vol. 498 Article 135441

We develop a novel cloud-function formalism describing the dynamical relationship between sensory-information processing in large-scale brain networks (supraliminal processing) and the content of the mental representation of an observed object. The formalism combines elements of neural field theory for large-scale neural activity with the spatial characteristics of perceived objects and their embedding in the environment ...

Added: October 2, 2026

Консервативные энтропийно и энергетически корректные разностные методы для одномерных квазигазодинамических систем уравнений

Zlotnik A., Математические заметки 2026 Т. 120 № 6 С. 1005–1009

Численным методам решения систем газодинамических уравнений посвящена обширная литература. Ранее было разработано и успешно апробировано специальное семейство симметричных по пространству консервативных разностных методов, основанных на предварительной кинетической, точнее, квазигазодинамической (КГД), регуляризации этих уравнений. Актуальной задачей является построение численных методов, которые обладают не только свойством консервативности по массе, импульсу и полной энергии, но и удовлетворяют условиям энтропийной ...

Added: October 1, 2026

On some arithmetic conditions of recurrent sequences modulo prime p

Vyugin I. V., Sashadhar D., Algebra and Number Theory 2026 P. 1–10

We study the K-Fibonacci sequence Fp modulo prime p. Cardinalities of sets |Fp+Fp| and |Fp⋅Fp| are estimated. We present the method of estimating doubling constant of some m-dimensional recurrent sets in Fp. ...

Added: October 1, 2026

Long-time behaviour of dynamical systems driven by bounded mixing noises

Kuksin S., Dynamical Systems 2026

We study the mixing properties of discrete-time and continuous-time dissipative dynamical systems driven by bounded mixing random forces. The continuous-time systems are reduced to discrete-time random dynamical systems generated by time-one maps, so that the main analysis is carried out in the discrete setting. We introduce a class of mixing random forcings whose regular conditional distributions with ...

Added: October 1, 2026

Markovian reduction and exponential mixing in total variation for random dynamical systems

Kuksin S., Shirikyan A., Journal of Dynamics and Differential Equations 2026 P. 1098–1100

The paper deals with the problem of large-time behaviour of trajectories for discrete-time dynamical systems driven by a random noise. Assuming that the phase space is finite-dimensional and compact, and the noise is a Markov process with a transition probability satisfying some regularity hypotheses, we prove that all the trajectories converge to a unique measure ...

Added: October 1, 2026

Bounds on the derivatives of the log-cumulative distribution function of the multivariate normal distribution

Potanin B., Dolgikh S., Statistics and Probability Letters 2027 Article 110984

We derive bounds on the gradient and Hessian of the log-CDF, ln F(x), of the multivariate normal distribution. These bounds scale linearly and quadratically in ‖x‖ , respectively, with constants depending only on the covariance matrix. We demonstrate the usefulness of these bounds by proving asymptotic normality of the maximum-likelihood estimator of the multivariate probit ...

Added: October 1, 2026

Asymptotics of Spectrum and Quantum Averages of the Hydrogen Atom in a Magnetic Field Near the Upper Boundaries of Spectral Clusters

A. V. Pereskokov, Journal of Mathematical Sciences 2026 Vol. 302 No. 4 P. 531–545

We consider the Zeeman effect problem for the hydrogen atom in a magnetic field using irreducible representations of the Karasev–Novikova algebra with quadratic commutation relations. We find the asymptotics of a series of eigenvalues and the corresponding asymptotic eigenfunctions near the upper boundaries of spectral clusters. ...

Added: October 1, 2026

Simultaneous approximation of the score function and its derivatives by deep neural networks

Yakovlev K., Puchkin N., Journal of Complexity 2026 Vol. 97

We present a theory for simultaneous approximation of the score function and its derivatives, enabling the handling of data distributions with low-dimensional structure and unbounded support. Our approximation error bounds match those in the literature while relying on assumptions that relax the usual bounded support requirement. Crucially, our bounds are free from the curse of ...

Added: September 30, 2026

Conservative Entropy and Energy Correct Difference Methods for One-Dimensional Quasi-Gasdynamic Systems of Equations

Zlotnik A., Mathematical notes 2026 Vol. 120 No. 6 P. 1174–1178

Numerical methods for solving systems of gas dynamic equations are the subject of a vast literature. A special family of spatially symmetric conservative difference methods based on preliminary kinetic, or quasi-gasdynamic (QGD), regularization of these equations was constructed and successfully tested. A pressing issue is the construction of numerical methods that are not only conservative ...

Added: September 30, 2026

О числе Пикара и размерности алгебраических однородных пространств

Beldiev I., Тимашёв Д. А., Алгебра и анализ 2026 Т. 38 № 5 С. 1–10

An algebraic variety X is called a homogeneous space if there exists a transitive regular action of an algebraic group on X. We prove inequalities between the dimension of a homogeneous space of a linear algebraic group and its Picard number. ...

Added: September 30, 2026

An archaic reference-free method to jointly infer Neanderthal and Denisovan introgressed segments in modern human genomes,

Planche L., Ilina A., Jay F. et al., Molecular Biology and Evolution 2026 Article 235

Admixture between populations is a common feature of human history. Admixture events introduce new genetic variation that can fuel evolution. Characterizing the significance of admixture events on the evolution of populations across various species is of great interest to evolutionary geneticists. Local Ancestry Inference (LAI) methods infer genetic ancestry of an individual at a particular ...

Added: September 30, 2026

Об асимптотических собственных значениях оператора типа Хартри вблизи границ спектральных кластеров.

А.В. Перескоков, В кн.: Современные методы теории краевых задач. Понтрягинские чтения XXXVII.: Воронеж: Издательский дом ВГУ, 2026. С. 251–252.

Рассматривается задача на собственные значения для возмущенного двумерного осциллятора. В качестве возмущения выступает интегральная нелинейность типа Хартри, потенциал самодействия в которой зависит от расстояния между точками и имеет одновременно кулоновскую и логарифмическую особенность. Найдены асимптотические собственные значения и асимптотические собственные функции вблизи границ спектральных кластеров, которые образуются около собственных значений невозмущенного оператора. Доказана новая формула ...

Added: June 25, 2026

Квазиклассическая асимптотика спектра трехмерного оператора типа Хартри вблизи верхних границ спектральных кластеров.

А.В. Перескоков, В кн.: Современные методы теории функций и смежные проблемы.: Издательский дом ВГУ, 2025. С. 267–268.

Найдены асимптотические собственные значения для нелинейного оператора типа Хартри в случае, когда потенциал самодействия является разностью кулоновского и экранированного кулоновского потенциалов. ...

Added: May 9, 2025

Об асимптотике спектра оператора типа Хартри, содержащего функцию Макдональда

А.В. Перескоков, В кн.: Современные методы теории краевых задач. Понтрягинские чтения — XXXV : материалы Международной Воронежской весенней математической школы (26 – 30 апреля 2024 г.).: Издательский дом ВГУ, 2024. С. 268–269.

Рассматривается задача на собственные значения для возмущенного двумерного осциллятора. В качестве возмущения выступает интегральная нелинейность типа Хартри, потенциал самодействия в которой содержит функцию Макдональда и зависит от расстояния между точками. Найдены асимптотические собственные значения и асимптотические собственные функции вблизи верхних границ спектральных кластеров. Около окружности, где локализовано решение, построено асимптотическое разложение. ...

Added: June 14, 2024

Asymptotics of the Spectrum of a Three-Dimensional Hartree Type Operator near Upper Boundaries of Spectral Clusters

A. V. Pereskokov, Journal of Mathematical Sciences 2024 Vol. 281 No. 4 P. 612–624

We consider the eigenproblem for a Hartree type operator with a small parameter at nonlineratiry, where the self-action potential is the difference between the Coulomb potential and the screened Coulomb potential. We find asymptotic eigenvalues and asymptotic eigenfunctions near the upper boundaries of spectral clusters which appear near the energy levels of an unperturbed operator. ...

Added: June 1, 2024

Асимптотические решения уравнения Хартри. Асимптотика самосогласованного потенциала

А.В. Перескоков, В кн.: Воронежская зимняя математическая школа С. Г. Крейна – 2024 : материалы международной Воронежской зимней математической школы, посвященной памяти В. П. Маслова (26–30 января 2024 г.).: Воронеж: Издательский дом ВГУ, 2024. С. 195–197.

Рассмотрена задача на собственные значения для оператора Хартри с малым параметром перед нелинейностью. Найдены асимптотические собственные значения и асимптотические собственные функции, локализованные вблизи сферы. Получены асимптотические разложения самосогласованного потенциала. ...

Added: May 3, 2024

Об асимптотических решениях двумерных уравнений типа Хартри, локализованных вблизи отрезков

А.В. Перескоков, В кн.: Современные методы теории краевых задач.: М.: МАКС Пресс, 2018. С. 170–171.

В работе найдены асимптотические решения двумерных уравнений типа Хартри, локализованные вблизи отрезков. ...

Added: December 15, 2023