Determinants of regular singular Sturm-Liouville operators

Abstract

We consider a regular singular Sturm-Liouville operator $L:=-\frac{d^2}{dx^2} + \frac{q(x)}{x^2 (1-x)^2}$ on the line segment $[0,1]$. We impose certain boundary conditions such that we obtain a semi-bounded self-adjoint operator. It is known that the $\zeta$-function of this operator $\zeta_L(s)=\sum_{\lambda\in\spec(L)\setminus\{0\}} \lambda^{-s}$ has a meromorphic continuation to the whole complex plane with 0 being a regular point. Then, according to Ray and Singer the $\zeta$-regularized determinant of $L$ is defined by $\detz(L):=\exp(-\zeta_L'(0)).$ In this paper we are 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, T. Dreyfus and H. Dym, and D. Burghelea, L. Friedlander and T. Kappeler. More precisely we prove the formula $\detz(L)=\frac{\pi W(\psi,\phi)} {2^{\nu_0+\nu_1} \Gamma(\nu_0+1)\Gamma(\nu_1+1)}.$ Here $\phi, \psi$ is a certain fundamental system of solutions for the homogeneous equation $Ly=0$, $W(\phi, \psi)$ denotes their Wronski determinant, and $\nu_0, \nu_1$ are numbers related to the characteristic roots of the regular singular points $0, 1$.