ABSTRACT
Motivated by problems of periodic motion in continuous media, such as the periodic flow of heat in a bar, Sturm and Liouville were led in 1836 to identify a class of problems in second-order differential equations that have inspired much of modern analysis and operator theory, and continue to do so. Their original formulation, already fairly general, called for the study of the nontrivial solutions of
(p(x)y′)′ + (λw(x) − q(x))y = 0, a ≤ x ≤ b, (1.1.1)
with homogeneous boundary conditions at x = a, x = b, of the form
Ay(a) +Bp(a)y′(a) = 0, Cy(b) +Dp(b)y′(b) = 0, (1.1.2)
where A,B are not both zero, and likewise C,D. Here all quantities are real, the prime (′) signifies d/dx, the functions p(x), w(x) are positive, and the interval (a, b) finite; we discuss these restrictions in more detail shortly. In their work, one meets, perhaps for the first time, the notion of an “eigenvalue” (or characteristic value), being a value of the parameter λ for which (1.1.1) has a non-trivial solution (the “eigenfunction”) satisfying the boundary conditions (1.1.2). The noteworthy propositions in these early papers include
λ 0 < λ
1 < λ
2 < . . . (1.1.3)
without finite limit-point,
(2) if y n (x) is the eigenfunction corresponding to λ
n , then y
n (x) has precisely
n zeros in the open interval (a, b), (Sturm oscillation theorem)
(3) the eigenfunctions are mutually orthogonal, in the sense∫ b a
ym(x)yn(x)w(x) dx = 0, m 6= n (1.1.4)
(4) the eigenfunctions form a “complete set,” in the sense that an “arbitrary” function f(x) from some general class can be expanded in some sense in the form
f(x) = ∞∑ n=0
cn yn(x), (1.1.5)
where the coefficients cn can be determined with the aid of the orthogonality relations (1.1.4),
(5) incidental lemmas, such as the Sturm comparison and separation theorems, estimates for eigenvalues, etc.
