Singular Sturm–Liouville problems whose coefficients depend rationally on the eigenvalue parameter

Abstract

Let − Dω(·, z) D+ q be a differential operator in L 2(0,∞) whose leading coefficient contains the eigenvalue parameter z. For the case that ω(·, z) has the particular form ω(t,z)=p(t)+c(t) 2/ z−r(t) , z∈ C⧹ R, and the coefficient functions satisfy certain local integrability conditions, it is shown that there is an analog for the usual limit-point/limit-circle classification. In the limit-point case mild sufficient conditions are given so that all but one of the Titchmarsh–Weyl coefficients belong to the so-called Kac subclass of Nevanlinna functions. An interpretation of the Titchmarsh–Weyl coefficients is given also in terms of an associated system of differential equations where the eigenvalue parameter appears linearly.