Self-adjoint Sturm-Liouville Problems Research Articles

An O(1) algorithm for the numerical evaluation of the Sturm-Liouville eigenvalues of the spheroidal wave functions of order zero

In addition to being the eigenfunctions of the restricted Fourier operator, the angular spheroidal wave functions of the first kind of order zero and nonnegative integer characteristic exponents are the solutions of a singular self-adjoint Sturm-Liouville problem. The running time of the standard algorithm for the numerical evaluation of their Sturm-Liouville eigenvalues grows with both bandlimit and characteristic exponent. Here, we describe a new approach whose running time is bounded independent of these parameters. Although the Sturm-Liouville eigenvalues are of little interest themselves, our algorithm is a component of a fast scheme for the numerical evaluation of the prolate spheroidal wave functions developed by one of the authors. We illustrate the performance of our method with numerical experiments.

Higher-order Sturm-Liouville problems with the Same eigenvalues

In this paper, we consider self-adjoint Sturm–Liouville problem SLP of higher-order. We define an equivalence relation between second- and higher-order SLP. Using the Darboux lemma and equivalence relation we obtain the closed form of a family of SLP which have the same eigenvalues. Also, some spectral properties of this family of Sturm–Liouville problems are investigated.

Winding in non-Hermitian systems

This paper extends the property of interlacing of the zeros of eigenfunctions in Hermitian systems to the topological property of winding number in non-Hermitian systems. Just as the number of nodes of each eigenfunction in a self-adjoint Sturm–Liouville problem are well-ordered, so too are the winding numbers of each eigenfunction of Hermitian and of unbroken -symmetric potentials. Varying a system back and forth past an exceptional point changes the windings of its eigenfunctions in a specific manner. Nonlinear, higher-dimensional, and general non-Hermitian systems also exhibit manifestations of these characteristics.

Geometric aspects of self-adjoint Sturm–Liouville problems

In this paper we use U(2), the group of 2 × 2 unitary matrices, to parametrize the space of all self-adjoint boundary conditions for a fixed Sturm–Liouville equation on the interval [0, 1]. The adjoint action of U(2) on itself naturally leads to a refined classification of self-adjoint boundary conditions – each adjoint orbit is a subclass of these boundary conditions. We give explicit parametrizations of those adjoint orbits of principal type, i.e. orbits diffeomorphic to the 2-sphere S2, and investigate the behaviour of the nth eigenvalue λnas a function on such orbits.

Dependence of eigenvalues on the boundary conditions of Sturm–Liouville problems with one singular endpoint

This paper is concerned with continuous and differentiable dependence of isolated eigenvalues on the boundary conditions of self-adjoint Sturm–Liouville problems with one singular endpoint. Locally continuous dependence of eigenvalues on the boundary conditions is proved. Especially, in the limit point case, the continuous dependence of isolated eigenvalues is shown by the relationships between Weyl–Titchmarsh m(λ)-function and the spectrum of the singular problems. Then continuous eigenvalue branches through each isolated eigenvalue over the space of boundary conditions are formed. It is rigorously shown that the eigenfunctions for the eigenvalues along a continuous simple eigenvalue branch can be continuously chosen with uniform bound in Lw2 norm. Its proof is different from that in the regular Sturm–Liouville problems. The derivative formulas of the continuous simple eigenvalue branch with respect to all the parameters in the boundary conditions are given, and thus its monotonicity with respect to some parameters is derived.

Self-adjoint Sturm-Liouville problems with an infinite number of boundary conditions

We study Sturm–Liouville (SL) problems on an infinite number of intervals, adjacent endpoints are linked by means of boundary conditions, and characterize the conditions which determine self-adjoint operators in a Hilbert space which is the direct sum of the spaces for each interval. These conditions can be regular or singular, separated or coupled. Furthermore, the inner products of the summand spaces may be multiples of the usual inner products with different spaces having different multiples. We also extend the GKN Theorem to cover the infinite number of intervals theory with modified inner products and discuss the connection between our characterization and the classical one with the usual inner products. Our results include the finite number of intervals case.

Self-adjoint Sturm-Liouville problems with discontinuous boundary conditions

We present a general framework for the study of self-adjoint SturmLiouville problems with discontinuous boundary conditions specified at interior points of the underlying interval. Some regular such conditions have been studied and are known by various names including transmission conditions, interface conditions, multi-point conditions, point interactions (in the Physics literature) etc. Using this framework we generate additional regular conditions and singular analogues of all these regular conditions. In the singular case the solutions and their (quasi) derivatives may have finite as well as infinite jump discontinuities.

Eigenvalues of limit-point Sturm–Liouville problems

The dependence of the eigenvalues of self-adjoint Sturm–Liouville problems on the boundary conditions when each endpoint is regular or in the limit-circle case is now, due to some surprisingly recent results, well understood. Here we study this dependence for singular problems with one endpoint in the limit-point case.

Matrix representations of Sturm–Liouville problems with eigenparameter-dependent boundary conditions

We show that a class of regular self-adjoint Sturm–Liouville problems with eigenparameter-dependent boundary conditions are equivalent to a certain class of matrix problems. Equivalent here means that they have exactly the same eigenvalues.

The index problem for eigenvalues for coupled boundary conditions and Fulton’s conjecture

This paper deals with regular self-adjoint Sturm–Liouville problems with coupled boundary conditions. The spectrum of the problems can be either bounded from below, or unbounded from both below and above. We consider the following question: given an eigenvalue, how to determine its index or indices efficiently? Using results on the level surfaces of the nth eigenvalue and certain inequalities among eigenvalues, the determination is converted into that of the index of the same eigenvalue for an appropriate separated boundary condition and hence can be achieved in terms of the Prufer angle. As an application of this complete solution of the index problem, we show that Fulton’s conjecture about such an index is true.

Limits of Sturm–Liouville eigenvalues when the interval shrinks to an end point

We investigate the behaviour of the eigenvalues of a self-adjoint Sturm–Liouville problem with a separated boundary condition when the interval of the problem shrinks to an end point. It is shown that all the eigenvalues, except possibly the first, approach limit. For the remaining choices of the boundary condition, several types of condition on the coefficient functions are given, so that the first eigenvalue has a finite or infinite limit and, when the limit is finite, an explicit expression for the limit is obtained. Moreover, numerous examples are presented to illustrate these results, and a construction is given to perturb the finite-limit case to the no-limit case.

Computing the indices of Sturm–Liouville eigenvalues for coupled boundary conditions (the EIGENIND-SLP codes)

The main purpose of the EIGENIND-SLP codes is to compute the indices of known eigenvalues of self-adjoint Sturm–Liouville problems with coupled boundary conditions (BCs). The spectrum of the problems can be unbounded from both below and above. Using some recent theoretical results, the computation is converted to that of the indices of the same eigenvalues for appropriate separated BCs, and is then carried out in terms of the Prüfer angle. The algorithm so generated and its implementation are discussed, and numerous examples are presented to illustrate the theoretical results and various aspects of the implementation.

Perturbations of Non Self-Adjoint Sturm-Liouville Problems, with Applications to Harmonic Oscillators

Perturbations of Non Self-Adjoint Sturm-Liouville Problems, with Applications to Harmonic Oscillators

Equality of multiplicities of a Sturm–Liouville eigenvalue

We give a new and unified proof of the fact that for any eigenvalue of a self-adjoint Sturm–Liouville problem with limit-circle end points, its analytic and geometric multiplicities are equal. This proof is geometric in nature and can be generalized to the case of high-order differential equations.

Geometric properties of the self-adjoint Sturm–Liouville problems

In this paper, geometric properties of the self-adjoint Sturm–Liouville problems are investigated. It is proved that for linear self-adjoint Sturm–Liouville problems, the eigenfunctions correspond exactly to the projections of the curvature lines on the energy functional surface with an appropriate metric and that the eigenvalues correspond exactly to the principal curvatures (at the origin) of the same energy functional surface.

Multiplicity of Sturm–Liouville eigenvalues

The geometric multiplicity of each eigenvalue of a self-adjoint Sturm–Liouville problem is equal to its algebraic multiplicity. This is true for regular problems and for singular problems with limit-circle endpoints, including the case when the leading coefficient changes sign.

Counting and computing eigenvalues of left-definite Sturm–Liouville problems

For left-definite regular self-adjoint Sturm–Liouville problems we construct a function N( λ) which counts the number of positive eigenvalues less than λ when λ is positive and the number of negative eigenvalues greater than λ when λ is negative. Based on this function we construct a code for the numerical computation of the eigenvalues and illustrate its use with examples. The boundary conditions can be separated or coupled. The code is available from www.mcs.le.ac.uk/~mmarletta/programs.html.

Left-Definite Sturm–Liouville Problems

Left-definite regular self-adjoint Sturm–Liouville problems, with either separated or coupled boundary conditions, are studied. We give an elementary proof of the existence of eigenvalues for these problems. For any fixed equation, we establish a sequence of inequalities among the eigenvalues for different boundary conditions and estimate the range of each eigenvalue as a function on the space of boundary conditions. Some of our results here yield an algorithm for numerically computing the eigenvalues of a left-definite problem with an arbitrary coupled boundary condition. Our inequalities imply that the well-known asymptotic formula for the eigenvalues in the separated case also holds in the coupled case. Moreover, we study the continuous and differentiable dependence of the eigenvalues of the general left-definite problem on all the parameters in its differential equation and boundary condition.

Algorithm 810: The SLEIGN2 Sturm-Liouville Code

The SLEIGN2 code is based on the ideas and methods of the original SLEIGN code of 1979. The main purpose of the SLEIGN2 code is to compute eigenvalues and eigenfunctions of regular and singular self-adjoint Sturm-Liouville problems, with both separated and coupled boundary conditions, and to approximate the continuous spectrum in the singular case. The code uses some new algorithms, which we describe, and has a driver program that offers a user-friendly interface. In this paper the algorithms and their implementations are discussed, and the class of problems to which each algorithm applied is identified.

Sturm–Liouville eigenproblems with an interior pole

The eigenvalues and eigenfunctions of self-adjoint Sturm–Liouville problems with a simple pole on the interior of the interval [A, B] are investigated. Three general theorems are proved and it is shown that as n→∞, the eigenfunctions more and more closely resemble those of an ordinary Sturm–Liouville problem and λn ∼−m2π2/(B−A)2, just as if there were no singularity. The low-order modes, however, differ drastically from those of a nonsingular eigenproblem in that (i) both eigenvalues and eigenfunctions are complex (despite the fact the problem is self-adjoint), (ii) the real and imaginary parts of the nth eigenfunction may both have ever-increasing numbers of interior zeros as B→∞, instead of just (n−1) zeros, and (iii) as B→∞, the eigenvalues for all small n may cluster about a common value in contrast to the widely separated eigenvalues of the corresponding nonsingular problem. These results are general, but in order to present quantitative solutions for the low-order modes, too, special attention is given to the particular case u″+(1/x−λ)u = 0, (1) with u(A) = u(B) = 0 where λ is the eigenvalue and A and B are of opposite signs. For small n, one can obtain the approximation λn∼exp[(1+31/2i)dn/(2B1/3)]/B, (2) where dn is the nth root of the Airy function Ai(−z). The imaginary part of (2) shows explicitly how profoundly the interior pole has modified the structure of the eigenproblem. The WKB method, which was used to derive (2), is shown to be accurate for all n. The WKB analysis is of some interest in and of itself. Although the number of WKB ’’transition’’ points is the same as for the half-century old quantum harmonic oscillator (two), the substitution of the interior pole for one of the turning points has a profound (and fascinating) impact on both the WKB formalism and the numerical results. Thus, although this problem was motivated by the physics of hydrodynamic waves, it is also an extension to both classical Sturm–Liouville theory and to the WKB treatment of eigenvalue problems.