A boundary value problem for a differential equation of second order

B. V. Verbitskii

Matematičeskie zametki, Tome 13 (1973) no. 3, pp. 373-384

Résumé

We find the spectrum and prove a theorem on the expansion of an arbitrary function satisfying certain smoothness conditions in terms of the root functions of a boundary value problem of the type \begin{gather*} -y''+q(x)+\frac a{x^2}y=\lambda y,\quad y(0)=0, \\ M(\lambda)y(a)+N(\lambda)y(b)=0, \end{gather*} where $0<a<b<\infty$, $a\ge0$, $M(\lambda)$ and $N(\lambda)$ are polynomials with complex coefficients, and $q(x)$ is a sufficiently smooth complex-valued function.