178 Chapter 2 The Heat Equation
is called aregular Sturm–Liouville problemif the following conditions are ful-
filled:
a.s(x),s′(x),q(x),andp(x)are continuous forl≤x≤r;
b. s(x)>0andp(x)>0 forl≤x≤r;
c. Theα’s andβ’s are nonnegative, andα 12 +α^22 >0,β 12 +β 22 >0.
d. The parameterλoccurs only where shown.Condition a and the first condition b guarantee that the differential equation
has solutions with continuous first and second derivatives. Notice thats(l)and
s(r)must both be positive (not zero). Condition c just says that there are two
boundary conditions:α 12 +α 22 =0onlyifα 1 =α 2 =0, which would be no
condition. The other requirements contribute to the desired properties in ways
that are not obvious.
We are now ready to state the theorems that contain necessary information
about eigenfunctions.
Theorem 1. The regular Sturm–Liouville problem has an infinite number of
eigenfunctionsφ 1 ,φ 2 ,...,eachcorrespondingtoadifferenteigenvalueλ^21 ,λ^22 ,....
If n=m, the eigenfunctionsφnandφmare orthogonal with weight function p(x):
∫r
lφn(x)φm(x)p(x)dx= 0 , n=m. The theorem is already proved, for the continuity of coefficients and eigen-
functions makes our previous calculations legitimate. It should be noted that
any constant multiple of an eigenfunction is also an eigenfunction; but aside
from a constant multiplier, the eigenfunctions of a Sturm–Liouville problem
are unique.
A number of other properties of the Sturm–Liouville problem are known.
We summarize a few here.
Theorem 2. (a)The regular Sturm–Liouville problem has an infinite number of
eigenvalues, andλ^2 n→∞as n→∞.
(b)Iftheeigenvaluesarenumberedinorder,λ^21 <λ^22 <···, then the eigen-
function corresponding toλ^2 nhas exactly n− 1 zeros in the interval l<x<r
(endpoints excluded).
(c)If q(x)≥ 0 andα 1 ,α 2 ,β 1 ,β 2 , are all greater than or equal to zero, then all
theeigenvaluesarenonnegative.
Examples.
- We note that the eigenvalue problems in Sections 3–6 of this chapter are
all regular Sturm–Liouville problems, as is the problem in Eqs. (1)–(3) of