C*-algebras of singular integral operators on the line related to singular Sturm-Liouville problems

Abstract

We study certain C*-algebras of singular integral operators on the line related to the second order ordinary differential operators Ho=-d/dx p(x) d/dx + q(x), with smooth coefficients and domain C o ∞ (ℝ) on L2(IR). Using Gelfand theory we find the structure of such algebras and deduce Fredholm criteria for related classes of ordinary differential operators of all orders. We give a complete description of some special cases including the case where p=l and where q≥1 is an even polynomial of arbitrary even degree.