zbMATH — the first resource for mathematics

Singular eigenvalue problems for second-order linear ordinary differential equations. (English) Zbl 0914.34021
Given the second-order Sturm-Liouville equation \[ (p(t)x')'+\lambda q(t)=0,\quad p(t)>0,\;q(t)>0,\tag{\(*\)} \] with a real-valued parameter \(\lambda \) and \(t\in [a,\infty)\), the authors characterize the set of \(\lambda \) for which the principal solution to \((*)\) satisfies \(x(a)=0\). This problem can be considered as a “singular” extension of the classical Sturm-Liouville eigenvalue problem on a compact interval \([a,b]\) with Dirichlet boundary conditions.
Similar to this regular case, there exists a sequence \(0<\lambda _0<\lambda _1<\dots <\lambda _n\to \infty \) and the corresponding eigenvalue functions \(u_n\), \(n=0,1,\dots ,\) have exactly \(n\) zeros in \((a,\infty)\).
An open problem concerning a possible relaxation of the restriction on coefficients \(p,q\) in main results of the paper is discussed.
Reviewer: O.Došlý (Brno)

34B24 Sturm-Liouville theory
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
Full Text: EuDML