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Asymptotic expansion of solutions of the 2nd order difference equations in an unbounded domain
Acta Applicandae Mathematicae. 2025. Vol. 195. Article 4.
Difference equations play a crucial role in a wide array of mathematical and physical tasks. In this article, we focus on the analysis of a second order linear homogeneous difference equation with smooth coefficients via WKB method. It is well-known that such equations exhibit two WKB solutions in a segment devoid of turning and singular points. We establish a theorem demonstrating the existence of these solutions in an unbounded domain under certain conditions regarding the smoothness and growth behavior of the coefficients at infinity. Furthermore, using this theorem, we derive the asymptotic expansion of Laguerre polynomials for large orders and values, yielding estimates that align with existing results.
Keywords: semiclassical approximationквазиклассическое приближениеLaguerre polynomialsdifference equationsразностные уравненияWKB methodполиномы Лагерра
Publication based on the results of:
Podinovskiy V. V., Нелюбин А. П., Автоматика и телемеханика 2026 № 8 С. 110–123
Для многокритериальных задач принятия решений по аналогии с качественной вероятностью введены понятия полной и частичной качественной важности как бинарных отношений, обладающих постулируемыми
свойствами. Предложено новое определение отношения нестрогого предпочтения на множестве вариантов решений, порождаемое качественной
важностью. Исследованы его свойства. Указаны аналитические правила,
позволяющие попарно сравнивать варианты по предпочтительности. Проведено сравнение новых отношений предпочтения с разработанными ранее для задач, ...
Added: September 9, 2026
Podinovskiy V. V., Нелюбин А. П., Автоматика и телемеханика 2026 № 7 С. 113–126
Рассматриваются задачи принятия решений, когда предпочтения оцениваются в порядковой шкале, а возможности реализации значений
неопределенного фактора описываются качественной вероятностью (полной или только частичной). Вводятся определения отношений предпочтения и безразличий на множестве стратегий. Предлагаются простые
решающие правила, позволяющие сравнивать стратегии по предпочтительности, и приводятся иллюстративные примеры. ...
Added: September 9, 2026
Chemical Papers 2026
Benzenoid hydrocarbons are structurally regular aromatic compounds, which makes them well suited for quantitative structure–property relationship (QSPR) studies. This study presents a QSPR analysis of nineteen benzenoid hydrocarbon species using degree-based topological co-indices. We developed a Python program to compute eleven degree-based topological co-indices from molecular graphs and evaluated their ability to explain variations in ...
Added: September 8, 2026
Glutsyuk A., / Series arXiv "math". 2026.
B.Josephson (Nobel Prize, 1973) predicted a tunnelling effect for a system of two superconductors separated by a narrow dielectric (such a system is called Josephson junction): existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by the family of differential equations on the 2-torus, dθdτ=1ω(cosθ+B+Acosτ), which is known as ...
Added: September 8, 2026
Glutsyuk A., / Series arXiv "math". 2026.
A planar dual billiard is a planar curve γ equipped with a family (σP)|P∈γ of projective involutions of the projective lines LP tangent to γ at P that fix P. A dual billiard is called rationally integrable, if there exists a rational function R(x,y) of two variables (called first integral) whose restriction to each tangent ...
Added: September 8, 2026
Alexandrov A., Glutsyuk A., Journal of Differential Equations 2026 Vol. 465 Article 114178
The overdamped Josephson junction in superconductivity theory can be modeled by the family of dynamical systems on the torus, which is known as the RSJ model. This family admits an equivalent description by a family of second-order differential equations: special double confluent Heun equations. In the present paper, we construct two new families of dynamical systems on ...
Added: September 8, 2026
Glutsyuk A., Inventiones Mathematicae 2026 Vol. 245 P. 977–1058
A caustic of a strictly convex planar bounded billiard is a smooth curve whose tangent lines are reflected from the billiard boundary to its tangent lines. The famous Birkhoff Conjecture, studied by many mathematicians, states that if the billiard boundary has an inner neighborhood foliated by closed caustics, then it is an ellipse. In the paper we study ...
Added: September 8, 2026
Prikhodko Artem, Kubrak D., Compositio Mathematica 2026 Vol. 162 No. 6 P. 1377–1438
In this follow-up paper we show that smooth Hodge-proper stacks over O𝐾 are ℚ𝑝-locally acyclic: namely the natural map between étale ℚ𝑝-cohomology of the algebraic and Raynaud generic fibers is an equivalence. This establishes the ℚ𝑝-case of general conjectures made in D. Kubrak and A. Prikhodko [p-adic Hodge theory for Artin stacks, Mem. Amer. Math. ...
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Ismailov A., Constructive Approximation 2026
The measure of the positivity set {x ∈ [0; 2π] | f (x) > 0} of a trigonometric polynomial f is bounded from below by the Motzkin density.
We generalize the bound to polynomials in several variables and almost periodic functions.
We then use these generalizations to extend known results on Taikov’s problem. ...
Added: September 7, 2026
Kucheryavyy P., Математические заметки 2026 Т. 2026 № 120 С. 380–401
В работе изучаются перестановки, возникающие при упорядочивании по возрастанию дробных долей произведений элементов фиксированной целочисленной последовательности на вещественный параметр. Исследуется количество различных перестановок, которые можно получить таким образом при изменении этого параметра от нуля до единицы. ...
Added: September 7, 2026
Осипов Д.В., Математический сборник 2026 Т. 217 № 9 С. 130–146
Изучаются законы взаимности, связанные с комплексными линейными расслоениями на расслоениях на ориентируемые окружности. В частности, доказывается следующий закон взаимности. Пусть B – комплексное многообразие и πi:Mi→B – расслоение на ориентируемые окружности, где индекс i пробегает конечное множество. Пусть Li и Ni – комплексные линейные расслоения на каждом многообразии Mi. Закон взаимности утверждает, что сумма всех элементов (πi)∗(c1(Li)∪c1(Ni)), где (πi)∗ – ...
Added: September 3, 2026
Basalaev A., Rarovskii A., Journal of Singularities 2026 Vol. 30 P. 61–80
Saito theory associates to an isolated singularity rich structure that plays an important role in mirror symmetry. In this note we construct Saito theory for A and D type Landau-Ginzburg orbifolds. Namely, for the pairs (f,G), where f defines an isolated singularity of A and D type and G is a group of symmetries of ...
Added: September 1, 2026
Rybakov M., Shkatov D., Journal of Logic and Computation 2026 Vol. 36 No. 6 Article exag026
We prove Pi-1-1-hardness, and thus lack of recursive axiomatizability, of constant-domain modal predicate logics defined by a class of Dedekind complete linear Kripke frames containing a frame with an infinitely increasing chain of worlds. The result holds even for the language with one unary predicate letter, one propositional letter, and two individual variables. ...
Added: September 1, 2026
Селянин Ф. И., Moscow Mathematical Journal 2026 Vol. 26 No. 2 P. 167–187
Minkowski mixed volume of n subpolytopes D1,…,Dn of a polytope P⊂Rn clearly does not exceed the normalized volume n!Vol(P). Equality holds if and only if the subpolytopes are interlaced, i.e., each proper face F⊊P intersects at least dim(F)+1 of the polytopes Di. Efficiently computing mixed volumes for more general collections of subpolytopes is crucial for estimating the complexity of numerically solving polynomial systems.
Motivated by relaxing the bound dim(F)+1 to dim(F), we ...
Added: August 31, 2026
Kazaryan M., Dunin-Barkowski P., Bychkov B. et al., International Mathematics Research Notices 2026 Vol. 14 Article rnag146
We prove a recent conjecture of the fourth named author with P. Norbury that states a system of universal polynomial relations among the kappa classes on the moduli spaces of algebraic curves. The proof involves localization and materialization analysis of the spin Gromov–Witten theory of the projective line and is dictated by Z 2 -equivariant ...
Added: August 31, 2026
A. V. Pereskokov, Journal of Mathematical Sciences 2025 Vol. 291 No. 4 P. 544–553
We study the eigenvalue problem for a perturbed resonance oscillator. We find asymptotic
expansions of coherent states of the algebra su(2). We show that the asymptotic
eigenfunctions are localized near a circle and construct an expansion of asymptotic eigenfunctions
near this circle. ...
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Vybornyi E., Rumiantseva S., Математические заметки 2024 Т. 116 № 6 С. 862–880
In this paper, we consider the problem of constructing a semiclassical asymptotic estimate of the splitting between a pair of close lower-lying energy levels for a quadratic operator defined on the irreducible representation of the Lie algebra su(1,1). In Darboux coordinates on the hyperboloid the Hamiltonian defines the landscape of a symmetric double well. It ...
Added: November 12, 2024
Danilov V., Rakhel M., ZAMM Zeitschrift für Angewandte Mathematik und Mechanik 2024 Vol. 104 No. 3 Article e202300491
In this paper, we propose a scheme for constructing the asymptotics of the fundamental solution of a linear degenerate parabolic equation with a small parameter. The asymptotics is constructed using the operator representation of the Dirac delta function and the non-oscillating Wentzel-Kramers-Brillouin (WKB) method. In addition, a method to justify the obtained asymptotics by proving the ...
Added: December 22, 2023
Lomonosov T., / Series arXiv "math". 2023. No. 2307.07182.
In this paper, an algebraic modification of the method of undetermined coefficients for solving nonhomogeneous linear stationary difference equations for quasipolynomial right-hand sides is proposed. Although the classical method of undetermined coefficients is well-known in both differential equations and difference equations case, its application in the difference equations case is severely limited. For example, it ...
Added: October 30, 2023
Vybornyi E., Rumiantseva S., Математические заметки 2022 Т. 112 № 5 С. 665–681
The paper considers the problem of constructing semiclassical asymptotics of tunnel splitting of the spectrum of an operator given on an irreducible representation of the Lie algebra su(1, 1). It is assumed that the operator is a quadratic function of the generators of the algebra. Coherent states and coherent unitary transformation are presented, which make ...
Added: November 7, 2022
Spiridonov V., Sarkissian G. A., / Series arXiv "math". 2021. No. arXiv:2105.15031.
General reduction of the elliptic hypergeometric equation to the level of complex hypergeometric functions is described. Derived equation is generalized to the Hamiltonian eigenvalue problem for new rational integrable N-body systems emerging from particular degenerations of the elliptic Ruijsenaars and van Diejen models. ...
Added: November 9, 2021
Булеков А. А., Journal of Physics: Conference Series 2021 Vol. 1740 Article 012069
The paper is devoted to the construction of spectral series and the estimation of the approximation accuracy for the operator of the Curie – Weiss model. In the course of work, the operator is reduced to a tridiagonal form in the subspace of the original space, then to a secondorder difference equation. The admissibility of ...
Added: June 9, 2021
Springer, 2020.
This edited volume gathers selected, peer-reviewed contributions presented at the fourth International Conference on Differential & Difference Equations Applications (ICDDEA), which was held in Lisbon, Portugal, in July 2019.
First organized in 2011, the ICDDEA conferences bring together mathematicians from various countries in order to promote cooperation in the field, with a particular focus on applications. ...
Added: November 1, 2020
Синцова К. А., Лялинов М. А., Wave Motion, 2018.
In this work we construct and discuss special solutions of a homogeneous problem for the Laplace
equation in a domain with the cone-shaped boundaries. The problem at hand is interpreted as that
describing oscillatory linear wave movement of a uid under gravity in such a domain. These solutions are found in terms of the Mellin transform and ...
Added: November 27, 2019