On conditions for the multiple expandability of functions in the root elements of a pencil of ordinary differential operators (Q1930810)

Given the linear ordinary differential equation \[ l(y)\equiv \sum_{\scriptstyle k_0+k_1\leq n\atop \scriptstyle k_0<n}\lambda ^{k_0}A^{(k_0,k_1)}(x)\frac{d^{k_1}y}{dx^{k_1}}-\lambda ^ny=0,\quad a<x<b, \] with complex eigenvalue parameter \(\lambda \) and with boundary conditions \[ U(y)\equiv\sum_{\scriptstyle k_0+k_1\leq n\atop\scriptstyle k_0<n}\lambda ^{k_0}\left\{\alpha ^{(k_0,k_1)} \left.\frac{d^{k_1}y}{dx^{k_1}}\right|_{x=a}+\beta ^{(k_0,k_1)} \left.\frac{d^{k_1}y}{dx^{k_1}}\right|_{x=b}\right\}=0. \] The author assumes a regularity condition which is a generalization of the standard Birkhoff regularity. Then it is shown that \(n\) integrable functions, \(n-1\) of which are in the domain of the differential operator, can be expanded simultaneously in terms of eigenfunctions and associated functions of the boundary eigenvalue problem. An explicit integral formula involving the Green function describes this expansion. This result is related to expansion theorems obtained in [\textit{J. D. Tamarkin}, M. Z. 27, 1--54 (1927; JFM 53.0419.02); \textit{G. D. Birkhoff} and \textit{R. E. Langer}, American Acad. Proc. 58, No. 2, 49--128 (1923; JFM 49.0723.01); \textit{M. V. Keldysh}, Dokl. Akad. Nauk SSSR, n. Ser. 77, 11--14 (1951; Zbl 0045.39402); \textit{M. A. Naimark}, Linear differential operators. I. New York: Frederick Ungar Publishing (1967; Zbl 0219.34001); II. (1968; Zbl 0227.34020); \textit{R. Mennicken} and \textit{M. Möller}, Non-self-adjoint boundary eingenvalue problems. Amsterdam: North-Holland (2003; Zbl 1033.34001)].