Invariant transformations for the Sturm-Liouville operator

In this paper, we consider the Sturm-Liouville operator on a finite interval. For particular boundary conditions, a group of invariant transformations that preserve the operator spectrum is constructed. This result allows us to reconsider some old problems for the Sturm-Liouville operator. In particular, the influence of the group of transformations on the inverse problem is discussed. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 25–54.     

Necessary Conditions for the Localization of the Spectrum of the Sturm-Liouville Problem on a Curve

We consider the Sturm-Liouville operator on a convex smooth curve lying in the complex plane and connecting the points 0 and 1. We prove that if the eigenvalues λ_k with large numbers are localized near a single ray, then this ray is the positive real semiaxis. Moreover, if the eigenvalues λ_k are numbered with algebraic multiplicities taken into account, then λ_k ∼ π · k as k → +∞.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 72–84.Original Russian Text Copyright © 2005 by Kh. K. Ishkin.     

Ambarzumian’s Theorem for a Sturm-Liouville Boundary Value Problem on a Star-Shaped Graph

Ambarzumian’s theorem describes the exceptional case in which the spectrum of a single Sturm-Liouville problem on a finite interval uniquely determines the potential. In this paper, an analog of Ambarzumian’s theorem is proved for the case of a Sturm-Liouville problem on a compact star-shaped graph. This case is also exceptional and corresponds to the Neumann boundary conditions at the pendant vertices and zero potentials on the edges.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 78–81, 2005Original Russian Text Copyright © by V. N. Pivovarchik     

On the conditions of solvability and the resolvent of a second-order deferential boundary operator in a space of vector functions

A boundary differential operator generated by the Sturm-Liouville differential expression with bounded operator potential and nonlocal boundary conditions is considered. The conditions for a considered operator to be a Fredholm and solvable operator are established and its resolvent is constructed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 517–524, April, 1995.     

Characterization of the spectrum of the Sturm-Liouville operator with irregular boundary conditions

We consider the spectral problem generated by the Sturm-Liouville equation with arbitrary complex-valued potential, q(x), ∈ L ₂(0, π) and irregular boundary conditions. We derive necessary and sufficient conditions for a set of complex numbers to be the spectrum of such an operator.     

Cauchy-type problem for an abstract differential equation with fractional derivatives

The uniform well-posedness of a Cauchy-type problem with two fractional derivatives and bounded operator A is proved. For an unbounded operator A we present a test for the uniform well-posedness of the problem under consideration consistent with the test for the uniform well-posedness of the Cauchy problem for an equation of second order.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 28–41.Original Russian Text Copyright © 2005 by A. V. Glushak.This revised version was published online in April 2005 with a corrected issue number.     

The Volterra Property of Some Problems with the Bitsadze-Samarskii-Type Conditions for a Mixed Parabolic-Hyperbolic Equation

We consider problems with the Bitsadze-Samarskii-type conditions for a mixed parabolic-hyperbolic equation with noncharacteristic type change curve. We prove theorems on the unique existence of regular and strong solutions and the Volterra property for the problems under consideration.Original Russian Text Copyright © 2005 Berdyshev A. S.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 500–510, May–June, 2005.     

On eigenfunction expansions for a nonlinear Sturm-Liouville operator with spectral-parameter dependent boundary conditions

We study a nonlinear eigenvalue problem for a Sturm-Liouville operator on the interval (0, 1). The boundary conditions posed at both endpoints of the interval depend on the spectral parameter. We prove that the problem has an eigenfunction system that is a basis in the space L _p (0, 1) for p > 1 and a Riesz basis for p = 2.     

Localization principle for expansions of generalized functions with respect to the eigenfunctions of the Sturm-Liouville operator on a finite interval

For linear methods of summation of the expansions of generalized functions into a series with respect to the eigenfunctions of a Sturm-Liouville operator one establishes conditions under which the Riemann localization principle holds.Translated from Ukrainskii Maternaticheskii Zhurnal, Vol. 43, No. 5, pp. 703–706, May, 1991.     

Stieltjes Sturm-Liouville equations: Eigenvalue dependence on problem parameters

We examine properties of eigenvalues and solutions to a 2n-dimensional Stieltjes Sturm-Liouville eigenvalue problem. Existence and uniqueness of a solution has been established previously. An earlier paper considered the corresponding initial value problem and established conditions which guarantee that solutions depend continuously on the coefficients [L.E. Battle, Solution dependence on problem parameters for initial value problems associated with the Stieltjes Sturm-Liouville equations, Electron. J. Differential Equations 2005 (2) (2005) 1-18]. Here, we find conditions which guarantee that the eigenvalues and solutions depend continuously on the coefficients, endpoints, and boundary data. For a simplified two-dimensional problem, we find conditions which guarantee the eigenvalues to be differentiable functions of the problem data.     

Bounds on Non-Real Eigenvalues of Indefinite Sturm-Liouville Problems

It is known since the early 20th century that regular indefinite Sturm-Liouville problems may possess non-real eigenvalues. However, finding bounds for this set in terms of the coefficients of the differential expression has remained an open problem until recently. In this note we prove a variant of a recent result in [1] on the bounds for the non-real eigenvalues of an indefinite Sturm-Liouville problem with Dirichlet boundary conditions. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Accuracy-enhancement scheme for the Sturm-Liouville equation

Enhanced-accuracy spline-difference schemes are constructed and analyzed for the one-dimensional Sturm-Liouville problem with piecewise-constant coefficients. Uniformmetric bounds are obtained for eigenvalues, eigenfunctions, and their derivatives. The results of numerical experiments using a test problem are reported.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 3–8, 1987.     

Asymptotic Expansion of Eigenvalues of the Laplace Operator in Domains with Singularly Perturbed Boundary

In this paper, we consider eigenvalue problems for the Laplace operator in three-dimensional domains with singularly perturbed boundary. Perturbations are generated by a complementary Dirichlet boundary condition on a small nonclosed surface inside the domain. The convergence and the asymptotic behavior of simple eigenvalues of the problem are considered.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 299–307.Original Russian Text Copyright © 2005 by M. I. Cherdantsev.     

Inverse problem of acoustic scattering for centrally symmetric finite objects in two-dimensional space

Scattering theory for the wave equation in two-dimensional space, perturbed by a finite function of a radial variable, integrable everywhere except, perhaps, the origin of coordinates, is considered from the point of view of the LaxPhillips scheme. The compression operator, related to the corresponding scattering problem, is considered. It is shown that this compression has one-dimensional defect subspaces, and its characteristic operator-function is a meromorphic function, whose zeros and poles coincide, respectively, with the corresponding values of a dissipative operator and its adjoint. The solution of the inverse scattering problem is obtained by reducing it to the inverse problem with two spectra for the singular self-adjoint Sturm-Liouville operator.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1649–1657, December, 1990.     

Convergence analysis of non-quadratic proximal methods for variational inequalities in Hilbert spaces

We consider a general approach for the convergence analysis of proximal-like methods for solving variational inequalities with maximal monotone operators in a Hilbert space. It proves to be that the conditions on the choice of a non-quadratic distance functional depend on the geometrical properties of the operator in the variational inequality, and –- in particular –- a standard assumption on the strict convexity of the kernel of the distance functional can be weakened if this operator possesses a certain `reserve of monotonicity'. A successive approximation of the `feasible set' is performed, and the arising auxiliary problems are solved approximately. Weak convergence of the proximal iterates to a solution of the original problem is proved.     

Matrix method for solving the Sturm-Liouville problem

We consider the application of the matrix method to constructing an approximate solution of the regular Sturm-Liouville problem with conditions of first, second, third, and mixed types. All these cases are reduced to a homogeneous algebraic system and its characteristic equation. A numerical example is given.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 43–49, 1985     

Stability of best rational Chebyshev approximation

In this paper we consider best Chebyshev approximation to continuous functions by generalized rational functions using an optimization theoretical approach introduced in [[5.]]. This general approach includes, in a unified way, usual, weighted, one-sided, unsymmetric, and also more general rational Chebychev approximation problems with side-conditions. We derive various continuity conditions for the optimal value, for the feasible set, and the optimal set of the corresponding optimization problem. From these results we derive conditions for the upper semicontinuity of the metric projection, which include some of the results of Werner [On the rational Tschebyscheff operator, Math. Z. 86 (1964), 317–326] and Cheney and Loeb [On the continuity of rational approximation operators, Arch. Rational Mech. Anal. 21 (1966), 391–401].     

Asymptotic Integration of Parabolic Problems with Large High-Frequency Summands

We develop the averaging method theory for parabolic problems with rapidly oscillating summands some of which are large, i.e., proportional to the square root of the frequency of oscillations. In this case the corresponding averaged problems do not coincide in general with those obtained by the traditional averaging, i.e., by formally averaging the summands of the initial problem (since the principal term of the asymptotic expansion of a solution to the latter problem is not in general a solution to the so-obtained problem). In this article we consider the question of time periodic solutions to the first boundary value problem for a semilinear parabolic equation of an arbitrary order 2k whose nonlinear terms, including the large, depend on the derivatives of the unknown up to the order k-1. We construct the averaged problem and the formal asymptotic expansion of a solution. When the large summands depend on the unknown rather than its derivatives we justify the averaging method and the complete asymptotic expansion of a solution.Original Russian Text Copyright © 2005 Levenshtam V. B.The author was supported by the Russian Foundation for Basic Research (Grant 01-01-00678) and the Program “ Universities of Russia” (UR.04.01.029).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 805–821, July–August, 2005.     

Reconstruction of Sturm-Liouville differential operators with singularities inside the interval

The inverse spectral problem for Sturm-Liouville differential operators on a finite interval is studied for an arbitrary and finite number of regular singular points inside the interval. A uniqueness theorem is proved; necessary and sufficient conditions and a procedure for the solution of the inverse problem are obtained.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 143–156, July, 1998.This research was supported by the Ministry of Education (KTsFE) under grant No. 96-1.7-4 and by the Russian Foundation for Basic Research under grant No. 97-01-00566.