Singular self-adjoint boundary value problems for the differential equation 𝐿𝑥=𝜆𝑀𝑥

Abstract

1. Introduction. Let L he a linear differential operator. Eigenfunction expansions for self-adjoint boundary value problems associated with the differential equation Zx=Xx on an infinite interval have been treated by various authors, beginning with Weyl [4] in 1910. Boundary value problems on a finite interval for differential equations of the form Lx=\Mx, where M is another linear differential operator whose order is less than that of L, have been discussed by Kamke [3]. The purpose of the first part of this paper is to improve the results of Kamke and to state them in a form suitable for the study of boundary value problems on an infinite interval. In the remainder of the paper the methods used by Coddington and Levinson [l ] for the equation Lx=\x are modified to study the equation Lx—\Mx on an infinite interval. 2. Self-adjoint boundary value problems on a finite interval. We consider the problem Lx=\Mx, Ux = 0 on the finite interval /, a^t^b. Here Lx = ZXo £>Wx(n-' Mx= YJ-o qi(t)x^m-l), are formally self-adjoint differential operators, with pi(t) and c7,(7) complex-valued functions with the properties that pi(t) is of class C-' on I, qi(t) is of class Cm_ 0 and K such that (Mu, u)^d(u, u) and (Lu, u)^K(u, u) ior all uED.