Green's functions for pairs of formally selfadjoint ordinary differential operators

Abstract

A spectral theory is deduced for differential eigenvalue problems related to a formally selfadjoint differential equation Su=λ Tu, where u is complex-valued and λ is the eigenvalue parameter. The equation or rather the differential operators S and T are considered on an arbitrary open interval of the real axis. The lower order operator T is assumed to have a positive definite Dirichlet integral which serves as scalar product in spectral theorems determined by symmetric boundary conditions. The theory is given in terms of ordered pairs u/\(\dot u\)of functions. Thus symmetric boundary conditions are certain subrelations of {u/\(\dot u\): Su=T\(\dot u\)}. If for instance T is the identity operator the boundary conditions are equally well described as conditions on u only. As far as the spectral theorem is concerned the method of the paper is easily transferred to the case when S instead of T has a positive definite Dirichlet integral. In the here considered case with T positive a kernel representation of the resolvent is deduced and used to prove the regularity of the elements of eigenspaces belonging to finite intervals of the spectral axis. The theory was worked out independently of the investigations by F.W. Schafke, A. Schneider and H.-D. Niesser of systems of first order equations to which it seems related in different respects.