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Том 241, № 5 (2019)
- Год: 2019
- Статей: 10
- URL: https://bakhtiniada.ru/1072-3374/issue/view/15024
Article
Identification of Nonseparated Boundary Conditions
Аннотация
We consider the problem of identification of nonseparated boundary conditions by five eigenvalues. Based on the Plücker conditions that appear in the problem of matrix recovery by its maximal-size minors, we construct the well-posedness set for this problem. We solve the problem of identification of the matrix of nonseparated boundary conditions in terms of the characteristic determinant of the corresponding spectral problem. The corresponding examples are presented.
507-517
Estimates of Initial Scales for Layers with Small Random Negative-Definite Perturbations
Аннотация
In this work, we consider the Schrödinger operator in a multi-dimensional layer with small random perturbations. Perturbations are distributed in periodicity cells of an arbitrarily chosen periodic lattice. To each cell, we put in correspondence a random variable; these random variables are independent and have the same distributions. Perturbations are described by the same abstract symmetric operator depending on the random variable multiplied by a global small parameter. We consider the case where the perturbations shift the bottom part of the spectrum of the unperturbed operator to the left by a quantity of order of the square of the small parameter. Under these conditions, we establish the main result, which is the estimate of initial scales. We also present particular examples that demonstrate the main result.
518-548
Convergence of Eigenfunctions of a Steklov-Type Problem in a Half-Strip with a Small Hole
Аннотация
We consider a Steklov-type problem for the Laplace operator in a half-strip containing a small hole with the Dirichlet conditions on the lateral boundaries and the boundary of the hole and the Steklov spectral condition on the base of the half-strip. We prove that eigenvalues of this problem vanish as the small parameter (the “diameter” of the hole) tends to zero.
549-555
On the Localization Conditions for the Spectrum of a Non-Self-Adjoint Sturm–Liouville Operator with Slowly Growing Potential
Аннотация
We consider the Sturm–Liouville operator T0 on the semi-axis (0,+∞) with the potential eiθq, where 0 < θ < π and q is a real-valued function that may have arbitrarily slow growth at infinity. This operator does not meet any condition of the Keldysh theorem: T0 is non-self-adjoint and its resolvent does not belong to the Neumann–Schatten class for any p < ∞. We find conditions for q and perturbations of V under which the localization or the asymptotics of its spectrum is preserved.
556-569
Calculation of Spectral Characteristics of Perturbed Self-Adjoint Operators by Methods of Regularized Traces
Аннотация
We discuss basic theoretical principles underlying new numerical methods of calculation of eigenvalues and eigenfunctions of discrete operators semi-bounded from below. We present algorithms of finding spectral characteristics by methods of regularized traces and examples related to certain spectral Sturm–Liouville problems.
570-588
On the Separation Property of Nonlinear Second-Order Differential Operators with Matrix Coefficients in Weighted Spaces
Аннотация
We prove the separation property for a certain class of nonlinear second-order differential operators with variable matrix coefficients in weighted spaces, that, in general, are not weak perturbations of linear operators.
589-595
On the Separation Property of the Sturm–Liouville Operator in Weighted Spaces of Multiplicators
Аннотация
In this paper, we prove the separation theorem for the Sturm–Liouville operator in terms of point multiplicators in weighted Sobolev spaces. The research method is based on local estimates on intervals of characteristic length.
596-604
Characteristic Properties of Scattering Data for Discontinuous Schrödinger Equations
Аннотация
In this paper, we discuss the inverse scattering problem to recover the potential from the scattering data of a class of Schrödinger equations with a nonlinear spectral parameter in the boundary condition. It turns out that for real-valued potential function q(x), the scattering data is defined as in the non-self-adjoint case: the scattering function, the nonreal singular values, and normalization polynomials. Characteristic properties of the spectral data are investigated. The solution of the problem is constructed by using the Gelfand–Levitan–Marchenko procedure. The uniqueness of the algorithm for the potential with given scattering data is proved.
605-613
Inverse Problems for Initial Conditions of the Mixed Problem for the Telegraph Equation
Аннотация
In this paper, we examine inverse problems for initial conditions for the wave and telegraph equations and state uniqueness criteria. Solutions of these problems are constructed in the series form. In the proof of uniform convergence of these series, the problem on small denominators appears. We prove estimates of small denominators separated from zero and obtain asymptotics that allow one to justify the convergence in the class of regular solutions.
622-645
On Asymptotics of Solutions to Some Linear Differential Equations
Аннотация
In this paper, we find the principal asymptotic term at infinity of a certain fundamental system of solutions to the equation l2n[y] = λy of order 2n, where l2n is the product of second-order linear differential expressions and λ is a fixed complex number. We assume that the coefficients of these differential expressions are not necessarily smooth but have a prescribed power growth at infinity. The asymptotic formulas obtained are applied for the problem on the defect index of differential operators in the case where l2n is a symmetric (formally self-adjoint) differential expression.
614-621