DETERMINANTS OF REGULAR SINGULAR STURM-LIOUVILLE OPERATORS

We consider a regular singular Sturm-Liouville operator L := - d<SUP>2 </SUP>/dx<SUP>2</SUP> + q(x)/x<SUP>2</SUP>(1 - x)<SUP>2 </SUP>on the l ine segment [0,1]. We impose certain boundary conditions such that we obtain a semibounded self-adjoint operator. It is known (cf. Theorem 1 .1 below) that the zeta-function of this operator [GRAPHICS] has a mer omorphic continuation to the whole complex plane with 0 being a regula r point. Then, according to [RS] the zeta-regularized determinant of L is defined by det(zeta)(L) := exp (-zeta'(L,)(0)). In this paper we a re going to express this determinant in terms of the solutions of the homogeneous differential equation Ly = 0 generalizing earlier work of S. LEVIT and U. SMILANSKY [LS], T. DREYFUS and H. DYM [DD], and D. BUR GHELEA, L. FRIEDLANDER and T. KAPPELER [BFK1, BFK2]. More precisely we prove the formula det(zeta)(L) = pi W(psi phi)/2(nu)0+(nu)1 Gamma(nu( 0)+1)(nu(1)+1). Here phi, psi is a certain fundamental system of solut ions for the homogeneous equation Ly = 0, W(phi, psi) denotes their Wr onski determinant, and nu(0), nu(1) are numbers related to the charact eristic roots of the regular singular points 0, 1.