C-SYMMETRIC SECOND ORDER DIFFERENTIAL OPERATORS

Abstract

We consider a C-Symmetric second order linear differential operator on a half interval or the real line. We determine the spectrum and construct the resolvent and m-function. In addition we analyze the resolvent and m-function near their poles. Under the conditions of Theorem 2.2 we prove the essential spectrum is empty, and the operator has a compact resolvent. Integral conditions on the operator coefficients are given in Theorem 3.4 for the operator to be Hilbert-Schmidt. These conditions are new even in the selfadjoint case. This analysis is based on asymptotic integration. A central role is played by the Titchmarsh-Weyl m-function which is defined by square integrable functions and not by a nesting circle analysis.