Singular Eigenvalue Problems for Second Order Linear Ordinary Differential Equations

We consider linear differential equations of the form (p(t)x′)′ + λq(t)x = 0 (p(t) > 0, q(t) > 0) (A) on an infinite interval [a,∞) and study the problem of finding those values of λ for which (A) has principal solutions x0(t;λ) vanishing at t = a. This problem may well be called a singular eigenvalue problem, since requiring x0(t;λ) to be a principal solution can be considered as a boundary condition at t = ∞. Similarly to the regular eigenvalue problems for (A) on compact intervals, we can prove a theorem asserting that there exists a sequence {λn} of eigenvalues such that 0 < λ0 < λ1 < · · · < λn < · · · , lim n→∞ λn = ∞, and the eigenfunction x0(t;λn) corresponding to λ = λn has exactly n zeros in (a,∞), n = 0, 1, 2, . . . . We also show that a similar situation holds for nonprincipal solutions of (A) under stronger assumptions on p(t) and q(t). AMS Subject Classification. 34B05, 34B24, 34C10