A dissipative singular Sturm-Liouville problem with a spectral parameter in the boundary condition

In the Hilbert space , we consider nonselfadjoint singular Sturm-Liouville boundary value problem (with two singular end points a and b) in limit-circle cases at a and b, and with a spectral parameter in the boundary condition. The approach is based on the use of the maximal dissipative operator, and the spectral analysis of this operator is adequate for boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Sturm-Liouville equation. On the basis of the results obtained regarding the theory of the characteristic function, we prove theorems on completeness of the system of eigenvectors and associated vectors of the maximal dissipative operator and Sturm-Liouville boundary value problem.     

Numerical solution of the inverse spectral problem for Bessel operators

We consider some inverse spectral problems associated with the singular Sturm-Liouville equation

     

Computing Eigenvalues of Singular Sturm-Liouville Problems

We describe a new algorithm to compute the eigenvalues of singular Sturm-Liouville problems with separated self-adjoint boundary conditions for both the limit-circle nonoscillatory and oscillatory cases. Also described is a numerical code implementing this algorithm and how it compares with SLEIGN. The latter is the only effective general purpose software available for the computation of the eigenvalues of singular Sturm-Liouville problems.     

The E-Functional for Some Pairs of Groups

We describe a new algorithm to compute the eigenvalues of singular Sturm-Liouville problems with separated self-adjoint boundary conditions for both the limit-circle nonoscillatory and oscillatory cases. Also described is a numerical code implementing this algorithm and how it compares with SLEIGN. The latter is the only effective general purpose software available for the computation of the eigenvalues of singular Sturm-Liouville problems.     

Half-inverse problem for diffusion operators on the finite interval

The potential function q(x) in the regular and singular Sturm-Liouville problem can be uniquely determined from two spectra. Inverse problem for diffusion operator given at the finite interval eigenvalues, normal numbers also on two spectra are solved. Half-inverse spectral problem for a Sturm-Liouville operator consists in reconstruction of this operator by its spectrum and half of the potential. In this study, by using the Hochstadt and Lieberman's method we show that if q(x) is prescribed on , then only one spectrum is sufficient to determine q(x) on the interval for diffusion operator.     

Posterior two-sided estimates for eigenvalues of the singular Sturm-Liouville problem

Based on three-point difference and variational-difference schemes for auxiliary nonsingular spectral problems providing for a two-sided approximation of eigenvalues of the singular Sturm-Liouville problem, posterior upper and lower estimates for eigenvalues of the input singular problem are obtained. Bibliography: 3 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 39–49.     

Eigenvalue asymptotics for Sturm-Liouville operators with singular potentials

We derive eigenvalue asymptotics for Sturm-Liouville operators with singular complex-valued potentials from the space , α∈[0,1], and Dirichlet or Neumann-Dirichlet boundary conditions. We also give application of the obtained results to the inverse spectral problem of recovering the potential from these two spectra.     

On the similarity of a <Emphasis Type="Italic">J</Emphasis>-nonnegative Sturm-Liouville operator to a self-adjoint operator

In terms of Weyl-Titchmarsh m-functions, we obtain a new necessary condition for an indefinite Sturm-Liouville operator to be similar to a self-adjoint operator. This condition is used to construct examples of J-nonnegative Sturm-Liouville operators with singular critical point zero.     

Pseudospectral methods for solving an equation of hypergeometric type with a perturbation

Almost all, regular or singular, Sturm-Liouville eigenvalue problems in the Schrödinger form

     

Singular cycles of vector fields on regular parts of the boundary of Morse-Smale systems

At the boundary of the class of Morse-Smale vector fields there are vector fields whose unique degenerate phenomena is a singular cycle. We first characterize and classify all singular cycles which contains only one degeneracy (thesimple singular cycles: ssc). Each of these cycles defines a condimension one submanifold of vector fields. For some ssc its codimension one submanifold is a regular part of the boundary of the Morse-Smale systems. We characterize those ssc that defines this type of submanifold. Our ambient space isn dimensional,n2.Supported by Fondecyt, Proyecto 1930863.     

Construction of a fundamental series of solutions of a pencil of matrices

Solution of spectral problems for a singular polynomial pencil of matrices D () of degree s1 and sizem×n is considered. Two algorithms for constructing polynomials solutions of pencils D () are considered: the first is a modification of an algorithm proposed earlier by one of the authors for determining polynomial solutions of a linear pencil; the second algorithm is based on other ideas and consists of two steps. At the first step a finite sequence of auxiliary pencils is constructed for each of which a basis of polynomial solutions of degree zero is found. At the second step the basis so constructed are rearranged into polynomial solutions of the original polynomial pencil D(). Both algorithms make it possible to find solutions of the original pencil in order of increasing degrees. For constructing a fundamental series of solutions of the pencil D() two new algorithms are proposed which work independently with either of the algorithms mentioned above for constructing polynomial solutions by rearranging them into linearly independent solutions of the pencil.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 139, pp. 74–93, 1984.     

Classical and vector sturm—liouville problems: recent advances in singular-point analysis and shooting-type algorithms

Significant advances have been made in the last year or two in algorithms and theory for Sturm—Liouville problems (SLPs). For the classical regular or singular SLP −(p(x)u′)′ + q(x)u = λw(x)u, a < x < b, we outline the algorithmic approaches of the recent library codes and what they can now routinely achieve.

For a library code, automatic treatment of singular problems is a must. New results are presented which clarify the effect of various numerical methods of handling a singular endpoint.

For the vector generalization −(P(x)u′)′+Q(x)u = λW(x)u where now u is a vector function of x, and P, Q, W are matrices, and for the corresponding higher-order vector self-adjoint problem, we outline the equally impressive advances in algorithms and theory.     

On the Neumann problem for the Sturm-Liouville equation with Cantor-type self-similar weight

The second and third boundary value problems for the Sturm-Liouville equation in which the weight function is the generalized derivative of a Cantor-type self-similar function are considered. The oscillation properties of the eigenfunctions of these problems are studied, and on the basis of this study, known asymptotics of their spectra are substantially refined. Namely, it is proved that the function s in the well-known formula $$N(\lambda ) = \lambda ^D \cdot [s(\ln \lambda ) + o(1)]$$ decomposes into the product of a decreasing exponential and a nondecreasing purely singular function (and, thereby, is not constant).     

Numerische Behandlung singulärer Sturm-Liouville-Probleme

Summary Generalizing the method of Wendroff [9] and using an estimate for the square integral of a normed eigenfunction outside a compact set, bounds are obtained for the eigenvalues of singular Sturm-Liouville problems from a finite difference method. The number of mesh points necessary to obtain. the accuracy behaves like ^–&frac; ln if tends to zero. Some numerical examples are given.     

Variational Principles for Eigenvalues of Self-adjoint Operator Functions

Variational principles for eigenvalues of certain functions whose values are possibly unbounded self-adjoint operators T() are proved. A generalised Rayleigh functional is used that assigns to a vector x a zero of the function T()x, x), where it is assumed that there exists at most one zero. Since there need not exist a zero for all x, an index shift may occur. Using this variational principle, eigenvalues of linear and quadratic polynomials and eigenvalues of block operator matrices in a gap of the essential spectrum are characterised. Moreover, applications are given to an elliptic eigenvalue problem with degenerate weight, Dirac operators, strings in a medium with a viscous friction, and a Sturm-Liouville problem that is rational in the eigenvalue parameter.     

Spectral analysis of singular ordinary differential operators with indefinite weights

In this paper we develop a perturbation approach to investigate spectral problems for singular ordinary differential operators with indefinite weight functions. We prove a general perturbation result on the local spectral properties of selfadjoint operators in Krein spaces which differ only by finitely many dimensions from the orthogonal sum of a fundamentally reducible operator and an operator with finitely many negative squares. This result is applied to singular indefinite Sturm-Liouville operators and higher order singular ordinary differential operators with indefinite weight functions.     

On Two-Dimensional Finite-Gap Potential Schroedinger and Dirac Operators with Singular Spectral Curves

We describe a wide class of two-dimensional potential Schroedinger and Dirac operators which are finite-gap at the zero energy level and whose spectral curves at this level are singular, in particular may have n-multiple points with n3.     

Transformations of diffusion and Schrödinger processes

Summary A transformation by means of a new type of multiplicative functionals is given, which is a generalization of Doob's space-time harmonic transformation, in the case of arbitrary non-harmonic function (t, x) which may vanish on a subset of [a, b]x^d. The transformation induces an additional (singular) drift term /, like in the case of Doob's space-time harmonic transformation. To handle the transformation, an integral equation of singular perturbations and a diffusion equation with singular potentials are discussed and the Feynman-Kac theorem is established for a class of singular potentials. The transformation is applied to Schrödinger processes which are defined following an idea of E. Schrödinger (1931).To commemorate the centenary of E. Schrödinger's birth (1887–1961)     

Inverse problem for Sturm-Liouville and hill equations

Summary We discuss the inverse Sturm-Liouville problem on a finite interval by the method of transformation kernel. The -function, the Fredholm determinant of the transformation kernel, is explicitly written down in terms of the spectral data, from which a very explicit representation formula for the potential is deduced, and well-posedness of the inverse problem is established. The above method is also applicated to the inverse problem for Hill equations, in particular to the isospectral problem. We obtain an analog of FIT formula and a regularity theorem.     

Computing eigenelements of Sturm-Liouville problems by using Daubechies wavelets

This work is our first step to get multiresolution approximation of eigenelements of Sturm-Liouville problems within bounded domain of varied nature. The formula for obtaining elements of representation of Sturm-Liouville operator involving polynomial coefficients in wavelet basis of Daubechies family have been derived in a form which can be readily used for their computations by a simple computer program. Estimates of errors for both the eigenvalues and eigenfunctions are also presented here. The proposed wavelet-Galerkin scheme based on scale functions and wavelets of Daubechies family having three or four vanishing moments of their wavelets has been applied to get approximate eigenelements of regular and singular Sturm-Liouville problems within bounded domain and compared with the exact or approximate results whenever available. From our study it appears that the proposed method is efficient and rapidly convergent in comparison to other approximation schemes based on variational method in Haar basis or finite difference methods studied by Bujurke et al. [39].