Sturm-Liouville Problem

the problem of finding nonzero solutions of the differential equation

(1) –[p(x)y’]’ + q(x)y = λy

that satisfy boundary conditions of the form

A₁y (a) + B₁y’ (a) = 0

A₂y (b) + B₂y’ (b) = 0

and of finding the values of the parameter λ for which these solutions exist. Such solutions are called eigenfunctions, and the corresponding values of λ are called eigenvalues. Under certain conditions on the coefficients p(x) and q(x), the Sturm-Liouville problem can be reduced to the consideration of the analogous problem for an equation of the form

(2) –y″ q(x)y = λy

The problem was first investigated between 1837 and 1841 by J. Liouville and J. C. F. Sturm.

The solution of some types of equations of mathematical physics by the Fourier method leads to the Sturm-Liouville problem. For example, the problem of the vibrations of a string of uniform density that is fastened at the ends leads to a Sturm-Liouville problem for the equation –y″ = λy with the boundary conditions y(0) = y(π) = 0. In this case there is an infinite sequence of values 1², 2²,..., n²,... to which the eigenfunctions sin nx, which form a complete orthogonal system of functions on the interval [0, π], correspond. A similar situation obtains in the general case, which arises, for example, in the study of the flow of heat in a heterogeneous bar. Here too, if the function q(x) in equation (2) is continuous and real on the interval [a, b] and A₁B₁, A₂, and B₂ are real numbers, then there exists an increasing sequence of real eigenvalues λ₁...,λ_n,... that tends to infinity. To each λ_n there corresponds an eigenfunction φ_n (x) that is defined to an accuracy of a constant factor and that has n zeros on the interval a < x < b. The functions φ_n(x) form on [a, b] a complete orthogonal system of functions; for equation (1) the eigenfunctions are orthogonal with respect to the weight function p(x).

The completeness of the system of eigenfunctions was proved by V. A. Steklov in 1896. In 1904, D. Hilbert, using the theory of linear integral equations, proved extremely general theorems on the expansion of functions in Fourier series with respect to the system φ_n(x). As n increases, the eigenvalues and eigenfunctions of the Sturm-Liouville problem for equation (2) tend to the eigenvalues and eigenfunctions for the equation –y″ = λy under the same boundary conditions. The majority of the orthogonal systems of functions encountered in mathematics, such as Legen-dre polynomials and Hermite polynomials, are systems of eigenfunctions of some Sturm-Liouville problems.

Also sometimes called a Sturm-Liouville problem is the boundary value problem of equation (1) under the more general boundary conditions

α_iy(a) + β_iy’ (a ) + γ_iy(b ) + δ_iy’ (b) = 0 i = 1, 2

where α_i, β_i, γ_i, and δ_i, are constants. The most important boundary conditions of this form are y(a) = y(b) and y′(a) = y′(b) (periodic conditions) and y(a) = –y(b) and y′(a) = –y’(b) (semi-periodic conditions).

Many problems of mathematical physics, such as the problem of the flow of heat in an infinite heterogeneous bar, lead to a Sturm-Liouville problem on a half axis or on the entire axis. In the first case, solutions of equation (2) that satisfy the condition A₁y(0) + B₁y′(0) = 0 are considered; instead of a sequence of eigenfunctions, we have the set of eigenfunctions φ(x, λ), which depend on the continuously changing parameter λ. Instead of an expansion in Fourier series, expansions of the following form are considered:

where ρ(λ) is some nondecreasing function. These expansions are analogous to the Fourier integral. Here

and

A similar situation occurs for the Sturm-Liouville problem on the entire axis.

For some problems of mathematical physics the inverse Sturm-Liouville problem, or the problem of reconstructing the differential equation from the function ρ(λ), is of great importance. This problem, which was posed in a special case by V. A. Ambartsu-mian and in a more general case by the Swedish mathematician G. Borg, was solved by M. G. Krein, I. M. Gel’fand, and B. M. Levitan.

The Sturm-Liouville problem also arises in some problems of quantum mechanics and the calculus of variations.