Inverse scattering problem for the Sturm-Liouville equation with spectral parameter in the discontinuity condition

We give a complete solution of the inverse scattering problem for the Sturm-Liouville equation with spectral parameter in the discontinuity condition in the absence of discrete spectrum.     

The Sturm-Liouville problem with a nonlocal boundary condition

In this paper, we consider the Sturm-Liouville problem with one classical and another nonlocal boundary condition. We investigate general properties of the characteristic function and spectrum for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues, analyze the dependence of the spectrum on parameters of the boundary condition, and describe the qualitative behavior of all eigenvalues subject to of the nonlocal boundary condition. Dedicated to N. S. Bakhvalov (1934–2005) Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 410–428, July–September, 2007.     

Asymptotic analysis of a singular Sturm-Liouville boundary value problem

Asymptotic expansions are given for the eigenvalues λ_n and eigenfunctions u_n of the following singular Sturm-Liouville problem with indefinite weight: $$\begin{gathered} - ((1 - x^2 )u'(x))' = \lambda xu(x) on ( - 1,1), \hfill \\ lim_{| x | \to 1} u(x) finite \hfill \\ \end{gathered} $$ This eigenvalue problem arises if one separates variables in a partial differential equation which describes electron scattering in a one-dimensional slab configuration. Asymptotic expansions of the normalization constants of the eigenfunctions are also given. The constants in these asymptotic expansions involve complete elliptic integrals. The asymptotic results are compared with the results of numerical calculations.     

On a classical problem with a complex-valued coefficient and the spectral parameter in a boundary condition

We consider a classical problem that arises when studying natural vibrations of a loaded string. We assume that the coefficient playing the role of a physical parameter can take complex values. We discuss the completeness, minimality, and basis property of the system of root functions.     

On the uniform convergence of the Fourier series for a spectral problem with squared spectral parameter in a boundary condition

We analyze the uniform convergence of the Fourier series expansions of Hölder functions in the system of eigenfunctions of a spectral problem with squared spectral parameter in a boundary condition. To this end, we first prove a theorem on the equiconvergence of such expansions with those in a well-known orthonormal basis.     

On the uniform convergence of the Fourier series for one spectral problem with a spectral parameter in a boundary condition

In this paper, we investigate the uniform convergence of the Fourier series expansions in terms of eigenfunctions for the spectral problem where λ is a spectral parameter, q(x) is a real‐valued continuous function on the interval [0,1], and a₁,b₀,b₁,c₁,d₀, and d₁ are real constants that satisfy the conditions Copyright © 2015 John Wiley & Sons, Ltd.     

Approximation of the eigenvalues of a singular Sturm-Liouville problem nonlinear in the parameter

Two auxiliary problems are solved on a truncated interval in the domain of definition with boundary conditions of first and second kind. The eigenvalues of the auxiliary problems provide a two-sided approximation of the eigenvalues of the original problem.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 11–16, 1990.     

On a problem with a boundary condition of the second kind,a complex-valued coefficient,and a spectral parameter in the other boundary condition

We study a classical problem that arises in the analysis of natural vibrations of a loaded string with a free endpoint. We assume that the coefficient occurring in the boundary condition of the third kind with a spectral parameter instead of a physical parameter can take complex values. We discuss the traditional aspects of the completeness, minimality, and basis property of the system of root functions. Special attention is paid to the structure of root subspaces.     

Global bifurcation results for some fourth-order nonlinear eigenvalue problem with a spectral parameter in the boundary condition

This paper is devoted to the study of global bifurcation from infinity of nontrivial solutions of a nonlinear eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter in the boundary condition. We prove the existence of two families of unbounded continua of nontrivial solutions to this problem, which emanate from bifurcation points in $ℝ\times \{\infty \}$ and possess oscillatory properties of eigenfunctions (and their derivatives) of the corresponding linear problem in some neighborhoods of these bifurcation points.     

On the multiple spectrum of a problem for the Bessel equation with spectral parameter in the boundary condition

A problem for the Bessel equation of zero order with complex physical and spectral parameters in the boundary condition is considered. We study whether the system of eigenfunctions has the basis property.     

Basis properties of a problem for the Laplace operator on the square with spectral parameter in a boundary condition

We consider a classical spectral problem that arises when studying the natural vibrations of a loaded rectangular membrane fixed on two sides, the load being distributed along one of the free sides. We study the completeness, minimality, and basis property of the system of eigenfunctions and establish conditions guaranteeing the equiconvergence of spectral expansions in this system and in a given basis.