Singular Sturm-Liouville Problems Research Articles

Computing eigenvalues of Sturm-Liouville systems of Bessel type

In this paper we shall develop a new method for the computation of eigenvalues of singular Sturm-Liouville problems of the Bessel type. This new method is based on the interpolation of a boundary function in Paley-Wiener spaces. Numerical results are provided to illustrate the method.

Singular nonlinear boundary value problems

We study nonlinear singular Sturm-Liouville bondary value prob lems associated with the operator , special cases of which come up in connection with the radial Laplace operator, and prove existence and uniqueness theorems under asymptotic non- resonance conditions on the slope of the nonlinear term with respect to the dependent variable. For the underlying linear problemsis allowed; nevertheless they have a pure point spectrum as in the classical case. In our treatment the Prüfer transformation plays an essential role. Our results apply in the case where Dirchlet's problem with radial data is considered in a ball, whereas this problem on an annulus leads to regular Sturm-Liouville problems.

Sturm—Liouville problems and discontinuous eigenvalues

If a Sturm—Liouville problem is given in an open interval of the real line, then regular boundary value problems can be considered on compact sub-intervals. For these regular problems, all with necessarily discrete spectra, the eigenvalues depend on both the end-points of the compact intervals, and upon the choice of the real separated boundary conditions at these end-points. These eigenvalues are not, in general, continuous functionsof the end-points and boundary conditions. The paper shows the surprising form of these discontinuities. The results have applications to the approximations of singular Sturm—Liouville problems by regular problems, and to the theoretical aspects of the Sleign2 Computer program.

Asymptotics of the Titchmarsh—Weyl m-coefficient for non-integrable potentials

Asymptotic formulae for the Titchmarsh—Weyl m-coefficient on rays in the complex λ-plane for the equation − y ″ + qy = λy whenthe potential is limit circle and non-oscillatory at x = 0 are obtained under assumptions slightly more general than xq(x) ∈ L1(0,c). The behaviour of q at the right end-point is arbitrary and may fall in either the limit-point or limit-circle case. A method of regularization of the equation is given that can be made to depend either on a solution of the equation for λ = 0 or more directly on an approximation to the solution in terms of q. This enables equivalent definitions of the m-coefficient to be given for the singular Sturm—Liouville problem associated with a singular limit-circle boundary condition, and its associatedregular Sturm—Liouville problem. As a consequence, it becomes possible to apply asymptotic results obtained by Atkinson for the regular problem in order to give asymptoticresults for the singular problem. Potentials of the form q(x) = C/xj, 1 ≤ j < 2, are included. In the case j = 1, an independent calculation of the limit-point m-coefficient over the range (0,∞), relying on Whittaker functions, verifies the main result.

Spectral approximations on the triangle

In this paper we describe a new family of polynomials which are eigenfunctions of a singular Sturm–Liouville problem on the triangle T 2 = { ( x , y ) : x ≥ 0 , y ≥ 0 , x + y 1 } . The polynomials are shown to be orthogonal over T 2 with respect to a unit weight function, and may be used in approximations which are exponentially convergent for functions which are infinitely smooth in T 2 . The zeros of the polynomials may be used in cubature formulae on T 2 .

The computation of spectral density functions for singular Sturm-Liouville problems involving simple continuous spectra

The software package SLEDGE has as one of its options the estimation of spectral density functions p(t) for a wide class of singular Strurm-Liouville problems. In this article the underlaying theory and implementation issues are discussed. Several examples exhibiting quite varied asymptotic behavior in p are presented.

The asymptotic form of the spectral function in Sturm–Liouville problems with a large potential like −xc(c ≦ 2)

A new formula is given for the spectral function in singular Sturm–Liouville problems where the potential is like – (const.)xc(0 < c ≦ 2). The formula is used to answer an open question concerning the asymptotic form of the spectral function.

Uniqueness of positive solutions for Sturm-Liouville boundary value problems

Sufficient conditions for the uniqueness of positive solutions of singular Sturm-Liouville boundary value problems (BVP) { ( E ) ( | u ′ | m − 2 u ′ ) ′ + f ( t , u , u ′ ) = 0 , in ( θ 1 , θ 2 ) , m ≥ 2 , ( B C ) { α 1 u ( θ 1 ) − β 1 u ′ ( θ 1 ) = 0 , α 2 u ( θ 2 ) + β 2 u ′ ( θ 2 ) = 0 , \begin{equation*} \begin {cases} (\mathrm E) (|u’|^{m-2}u’)’+f(t,u,u’)=0,\quad \text {in} (\theta _1,\theta _2),m\ge 2,\ (\mathrm {BC})\begin {cases} \alpha _1u(\theta _1)-\beta _1u’(\theta _1)=0,\ \alpha _2u(\theta _2)+\beta _2u’(\theta _2)=0, \end{cases} \end{cases} \tag {BVP} \end{equation*} where α i , β i ≥ 0 \alpha _i,\beta _i\ge 0 and α i 2 + β i 2 ≠ 0 \alpha _i^2+\beta _i^2\not =0 ( i = 1 , 2 ) (i=1,2) , are established.

Maximum and anti-maximum principles for singular Sturm–Liouville problems

The maximum and anti-maximum principles are extended to the case of eigenvalue Sturm–Liouville problemswith boundary conditions of Dirichlet type (if possible) on a bounded interval [a, b]. The function r is assumed to be continuous and > 0 on ]a, b[, but the function 1/r is not necessarily integrable on [a, b]. The conditions on the functions p, m and h depend on the integrability or nonintegrability of 1/r on [a, c] and/or [c, b], for some c ∈ ]a, b[. The weight function m is not necessarily of constant sign.

On the asymptotic distribution of the eigenvalues of singular Sturm-Liouville problems with an indefinite weight function

We investigate spectrum and asymptotic distribution of the eigenvalues of certain singular Sturm-Liouville eigenvalue problems on the half-line having a weight function that changes sign.

Regular and singular Sturm–Liouville problems with coupled boundary conditions

Eigenvalues of both regular and singular Sturm–Liouville (S–L) problems with general coupled self-adjoint boundary conditions are characterised. This characterisation, although elementary, appears to be new even in the regular case. The singular characterisation is an exact parallel of the regular one and reduces to it. One application yields inequalities among the eigenvalues of different coupled boundary conditions. This is a far-reaching extension, even in the regular case, of the well-known relationship among the periodic and semiperiodic eigenvalues.

An Asymptotic Numerical Method for a Class of Singular Sturm–Liouville Problems

A class of Sturm–Liouville problems has been discovered for which the method of approximating the coefficients by piecewise constant functions has considerable difficulty. We characterize this class, give examples, and derive a modification of the method based on local asymptotic approximations which surmounts the difficulties.

SELF-ADJOINTNESS FOR THE WEYL PROBLEM UNDER AN ENERGY NORM

Abstract Since 1910 when H. Weyl first successfully attacked the singular Sturm-Liouville problem associated with a vast literature has arisen for “right definite” problems, lying in Hilbert spaces generated by the coefficient on the right side of the homogeneous differential equation. Less well examined is a similar theory for operators in a “left definite” setting, generated by the coefficients on the left side of the homogeneous differential equation. In this article we show that under favorable circumstances the left definite theory can be imbedded within the right definite, and will inherit many of the right definite properties.

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.

Hearing Point Masses in a String

The spectral and inverse spectral theory for certain singular Sturm–Liouville problems is developed. These boundary value problems arise when considering the wave equation corresponding to a string with finitely many embedded point masses. These singular eigenvalue equations,their solutions, and the associated Hilbert space operators are constructed as limits of regular problems. The eigenfunctions of the singular problem are shown to be solutions of a regular eigenvalue problem with interior point conditions.Expressions describing the distribution of large eigenvalues are developed. Algorithms are given for extracting information about the singularities from eigenvalues corresponding to one or two sets of boundary conditions. In the generic case a single spectrum determines the (unordered) set of lengths of the intervals separating the singularities.

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 an eigenvalue problem arising in the study of the stability of ocean currents

We study a singular Sturm-Liouville eigenvalue problem which arises in studying the stability of rotating shallow-water shear flows on an equatorial β-plane. Earlier studies have shown that the problem has only stable eigenvalues if a certain parameter φ exceeds 3/4. On the other hand, numerical studies have found unstable eigenvalues if φ is sufficiently small. In this note, we give an analytical proof that the bound of 3/4 is indeed sharp, i.e. there are instabilities for any φ < 3/4.

PARALLEL COMPUTATION OF STURM-LIOUVILLE SPECTRAL DENSITY FUNCTIONS∗ †

The estimation of spectral density functions for singular Sturm-Liouville problems having continuous spectrum is a computationally intensive task. In this paper we adapt the algorithm from the software package SLEDGE, which was written for serial machines, to a hypcrcube environment.

Singular sturm liouville problems and existence of solutions to singular nonlinear boundary value problems

Singular sturm liouville problems and existence of solutions to singular nonlinear boundary value problems

Regular approximations of singular Sturm-Liouville problems

Given any self-adjoint realization S of a singular Sturm-Liouville (S-L) problem, it is possible to construct a sequence {Sr{ of regular S-L problems with the properties