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Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika

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No 12 (2024)
3-11 121
Abstract

We consider the Schrödinger operator $H(\mathbf{k})=H_0(\mathbf{k})-V, \,\, \mathbf{k}\in \mathbb{T}^2,$ associated with a system of two particles on a two-dimensional lattice. It is shown that the subspaces of even as well as odd functions are invariant under operator $H(\mathbf{k}).$ The sets of quasimomenta $\mathcal{K}(1),$ $\mathcal{K}(2)$ and the class of potentials $\mathrm{P}(1),$ $\mathrm{P}(2)$ are described, for which the operator $H(\mathbf{k})$ has infinite number of eigenvalues $z_n(\mathbf{k}),\, n\in \mathbb{Z}_+$, for $\mathbf{k}\in \mathcal{K}(j),\,\, \hat{v}\in \mathrm{P}(j)$. The explicit form of $z_n(\mathbf{k})$ and the rate of convergence of the sequence $z_n(\mathbf{k})$ to the bottom of the essential spectrum are found.

12-19 100
Abstract

The article considers a boundary value problem with integral boundary conditions for one nonlinear second-order functional differential equation. Using Green’s function, the boundary value problem is reduced to the equivalent nonlinear Hammerstein integral equation. Next, having identified the necessary properties of Green’s function, we prove that the Hammerstein operator contracts the corresponding cone. The last circumstance, by virtue of the well-known Krasnoselsky theorem, guarantees the existence of at least one positive solution to the boundary value problem. Using a priori estimates and the principle of compressed mappings, sufficient conditions for the uniqueness of a positive solution were obtained. At the end of the article, there is a non-trivial example illustrating the results obtained here.

20-37 89
Abstract

Jacobi polynomial $P_{n}^{\left( \alpha ,\beta \right)}(x)$ is a well-known orthogonal polynomial. In the present work, several new properties of generalized Jacobi polynomial $P_{n,\tau}^{\left( \alpha ,\gamma,\beta \right)}(x)$ (Waghela D., Rao S.B. A Note on Sequence of Functions associated with the Generalized Jacobi polynomial, Researches Math. 31 (2), 1-18 (2023)) and its special case $P_{n}^{\left( \alpha ,\gamma,\beta \right)}(x)$ have been studied, which along with different representations of the said generalization includes crucial orthogonality property, generating function, results involving integral representation, differentiation of generalized Jacobi polynomial; also many well-known transformations of this generalized polynomial have been obtained.

38-43 50
Abstract

Let $D$ be a square with the boundary $\Gamma$. A four-element linear sum-difference equation is considered in the class of functions that are holomorphic outside $D$ and vanish at infinity. The coefficients of the equation and the free term are holomorphic in $D$. The solution is sought in the form of a Cauchy-type integral over $\Gamma$ with unknown density. Its boundary values satisfy the Hölder condition on any compact set in $\Gamma$ that does not contain vertices. At most, logarithmic singularities are allowed at the vertices. To regularize the equation on $\Gamma$, a piecewise linear Carleman shift is introduced, which changes the orientation and still has fixed points. It is continuous at the vertices, but its derivatives are discontinuous at them. The regularization of uranium was carried out and the condition for its equivalence was found. Various applications and generalizations are indicated.

44-56 140
Abstract

This paper considers the problem of constructing the Fourier transform of a harrow-shaped function to determine a discrete analog of the differential operator, which is used in constructing optimal quadrature formulas in L. Hörmander space. In addition, the problem of constructing a discrete analog of a specific operator in a particular case is considered.

57-70 115
Abstract

A problem with the Bitsadze-Samarskii and Frankl conditions for a mixed-type equation in a domain whose elliptic part is the first quadrant of the plane, and the hyperbolic part is the characteristic triangle is studied. The correctness of the formulated problem is proved.

71-84 94
Abstract

For linear systems of differential equations with delay subject to generalized influence, a formalization of the concept of Highers-Ulam-Rassias stability is proposed. The cases are considered when the system has a single reaction to a generalized impact and when the system's reaction is not unique. Sufficient conditions for such stability are established for the systems of differential equations under consideration.

85-93 136
Abstract

The aim of this paper is to present the results on the Hyers–Ulam–Rassias stability and the Hyers–Ulam stability for Bernoulli's differential equation. The argument makes use of a fixed point approach. Some examples are given to illustrate the main results.

94-100 102
Abstract

Surface periodic waves of infinite depth are investigated. The boundary value problem is formulated in the parametric plane with respect to the Zhukovsky function. By making use of the discrete Fourier transform, the problem is reduced to a finite system of nonlinear transcendental equations. It is shown that with an increase in the steepness of the waves, an inner solution is formed near the crest, and under the corresponding scaling of the sought function this solution is independent of the steepness. It is shown that the numerical reproduction of the inner solution is a key factor for accurate calculations of the almost-highest gravity waves.

101-108 120
Abstract

A two-dimensional hierarchical lattice is considered, in which an elementary cell is represented by the vertices of a square. In the generalized hierarchical model, the distance between opposite vertices of a square differs from the distance between neighboring vertices and is a parameter of the new model. At each vertex of the lattice, the field is defined by a set of 4 generators of the Grassmann algebra. The Hamiltonian of the field is described by the interaction of the 4th degree. The transformation of the renormalization group in the space of coupling coefficients defining this interaction is defined as a nonlinear mapping. All branches of fixed points of this mapping are described.



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ISSN 0021-3446 (Print)
ISSN 2076-4626 (Online)