Elgenfunction Expansions Associated with Second Order Differential Equations

I

THE STURM-LIOUVILLE EXPANSION

1.1. Introduction. Let L denote a linear operator operating on a function y = y(x). Consider the equation

where λ is a number. A function which satisfies this equation and also certain boundary conditions (e.g. which vanishes at x = a and x = b) is called an eigenfunction. The corresponding value of λ is called an eigenvalue. Thus if ψn(x) is an eigenfunction corresponding to an eigenvalue λn,

The object of this book is to study the operator

where q(x) is a given function of x defined over some given interval (a, b). In this case y satisfies the second-order differential equation

and ψn(x) satisfies

If we take this and the corresponding equation with m instead of n, multiply by ψm(x), ψn(x) respectively, and subtract, we obtain

Hence

if ψm(x) and ψn(x) both vanish at x = a and x = b (or satisfy a more general condition of the same kind). If λm ≠ λn, it follows that

By multiplying if necessary by a constant we can arrange that

The functions ψn(x) then form a normal orthogonal set.

Our main problem is to determine under what conditions an arbitrary function f(x) can be expanded in terms of such functions, in the manner of an ordinary Fourier series. If this is possible, and the expansion is

then on multiplying by ψm(x) and integrating over (a, b), we obtain formally

In some cases the eigenvalues are not discrete points, but form a continuous range, say, for example, over (0, ∞). The expansion then takes the form

All this has its simplest illustration in the case of ordinary Fourier series. Suppose, for example, that q(x) = 0, and that the interval considered is (0, π). The solution of (1.1.4) which vanishes at x = 0 is then y = sin(x√λ). This vanishes at x = π if and only if λ = n², where n is an integer. These then are the eigenvalues, and the corresponding eigenfunctions are the functions sin nx. That an arbitrary function can be expanded in terms of these functions is the familiar theorem on Fourier’s sine series.

1.2. An argument which has sometimes been used to suggest the validity of the above expansions runs as follows. Consider the partial differential equation

where f = f(x, t). If f(x, t) is given for one value of t, it is fixed by this equation for a slightly greater value of t. Thus we should expect to have one solution, and only one, for any given initial value of f(x, t), i.e. for f(x, t) equal to an arbitrary f(x) when t = 0. Suppose now that the solution can be expressed as a Fourier integral,

where

Substituting in (1.2.1), we obtain

Since this holds for all values of t, we can equate the coefficients of e–iλt, which gives

Thus F(x, λ) is an eigenfunction of the operator L belonging to the eigenvalue λ. Now putting t = 0 in (1.2.2), we obtain

which gives an expression for the arbitrary function f(x) in terms of eigenfunctions. If f(x, t) were expressible in a Fourier series instead of an integral, we should obtain similarly a series expansion.

The difficulty of justifying directly an argument such as the above is obvious.

1.3. The argument assumes, for one thing, that f(x, t) is small as t → ∞, since this is required for the Fourier integral formula (1.2.2), (1.2.3) to hold. However, a more general form of the formula is as follows. Let

The inverse formula is then

where c > 0, c′ < 0. Using this and (1.2.1), we obtain formally

if f(x, t) reduces to f(x) when t = 0. Similarly

The method to be employed is therefore as follows. We construct the solutions F+ and F– of (1.3.4) and (1.3.5) which satisfy given boundary conditions. Then (1.3.3), with t = 0, gives

In the simplest cases F–(x, λ) is found to be minus the analytic continuation of F+(x, λ) across the real axis. Writing

(1.3.4) becomes

and (1.3.6) becomes

The expansion is then obtained from the calculus of residues, the terms of the series being the residues at the poles of Φ(x, λ).

In any case, since (1.3.4) and (1.3.5) differ only in having the sign of i changed, the two terms in (1.3.6) are conjugate. Hence we also have

The expansion formula is obtained from this by making c → 0.

The above argument indicates in particular that λ must be treated as a complex variable; but the analysis is of course still purely formal.

1.4. In the particular case in which the operator L is given by (1.1.3), (1.3.8) is the second-order differential equation

In this case the function Φ(x, λ) can be expressed in terms of the solutions of (1.1.4). Let Wx( , ψ) or W( , ψ) denote the Wronskian W( , ψ) = (x)ψ′(x)– ′(x)ψ(x) of two functions and ψ. Now let (x, λ) and ψ(x, λ) be two solutions of (1.1.4) such that W( , ψ) = 1. Then a solution of (1.4.1) is

as is easily verified by differentiating twice.

Another starting-point for the theory, which (in the case of (1.1.3)) avoids an appeal to Fourier integrals, is as follows. Suppose the theory already established, and consider the properties of the function

This gives

This is (1.4.1) again, so that we are again led to consider solutions of this differential equation. If we can solve it, (1.4.3) then indicates that the terms of the expansion of f(x) will be the residues at the poles of Φ(x, λ).

Our general method consists of defining Φ(x, λ) by (1.4.2); integration round a large contour in the complex λ-plane then gives the value f(x), and the singularities of Φ(x, λ) on the real axis give a series or an integral expansion as the case may be.

1.5. The Sturm–Liouville expansion. We shall suppose throughout that q(x) is a real function of x, continuous at all points interior to the interval (a, b) considered. In the classical Sturm–Liouville case (a, b) is a finite interval, and q(x) also tends to finite limits as x → a and x → b.

The general theorem on the existence of solutions of (1.1.4) is as follows.

THEOREM 1.5. If q(x) satisfies the above conditions, and α is given, the equation (1.1.4) has a solution (x) (a ≤ x ≤ b) such that

For each x, (x) is an integral function of λ.

Let

and for n = 1, 2,...,

Let

Then

For n ≥ 1,

Hence

and so generally

Hence the series

converges, uniformly with respect to λ if |λ| ≤ N, and with respect to x over a ≤ x ≤ b. Since for n ≥ 2

the first and second differentiated series also converge uniformly with respect to x. Hence

so that (x) satisfies (1.1.4). It also clearly satisfies the boundary conditions.

1.6. Now let (x, λ), X(x, λ) be the solutions of (1.1.4) such that

Then

Hence W( , X) is independent of x, and so is a function of λ only say ω(λ). It is clear from the above theorem that it is an integra function of λ.

Let

It is at once verified by differentiation that Φ(x, λ) satisfies (1.4.1), and also the boundary conditions

for all values of λ.

Suppose that the only zeros of ω(λ) are simple zeros λ0, λ1,... on the real axis. Then the Wronskian of