Basic Properties of a Sturm-Liouville Eigenvalue Problem

It is surprising how much useful information one can infer about the eigenvalues and eigenfunctions of a S-L problem without actually solving the differential equation explicitly. Thus from very general and simple considerations we shall discover that the eigenvalues are real, are discrete if the domain is finite, have a lowest member, increase without limit, and that the corresponding eigenfunctions are orthogonal to each other, oscillate, oscillate more rapidly the larger the eigenvalue, to mention just a few pieces of useful information.

In practice this kind of information is quite often the primary thing of interest. In other words, the philosophy quite often is that one verifies that a certain system is of the S-L types, thus having at one's immediate disposal a concomitant list of properties of the system, properties whose qualitative nature is quite sufficient to answer the questions one had about the system in the first place.

As promised, we shall develop these and other properties by means of a collection of theorems. But before doing so, we remind ourselves about what is meant by a ``Sturm-Liouville system'', by a ``solution'', and by ``orthogonality''. The Sturm-Liouville system we shall consider consists of (i) the S-L differential equation

$$\left[ \frac{d}{dx} p(x)\frac{d}{dx} +q(x)+\lambda \rho (x)\right] u(x) = 0\,,$$ (310)

where $q$ , $\rho$ , $p$ , and $p'$ are continuous and $\rho$ and $p$ are positive definite functions on the open interval $(a,b)$ together with (ii) the boundary conditions

$$1. \alpha u(a)+\alpha ' u'(a)=0$$ (311)

$$2. \beta u(b)+\beta ' u'(b)=0$$

where the given constants $\alpha$ , $\alpha'$ , $\beta$ and $\beta'$ are independent of the parameter $\lambda$ .

Corresponding to an eigenvalue of this S-L system, an eigenfunction $u_n(x)$ is understood to be that solution which is ``regular'', i.e.,

$$u_n(x) ~\hbox{and}~ \frac{du_n(x)}{dx}~\hbox{are}~continuous~$$

and hence finite, on the closed interval $[a,b]$ . In particular, an eigenfunction must not have any finite jump discontinuities anywhere in $[a,b]$ .