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 $ζ$-function of this operator $ζ_L(s)=\sum_{λ\in\spec(L)\setminus\{0\}} λ^{-s}$ has a meromorphic continuation to the whole complex plane with 0 being a regular point. Then, according to Ray and Singer the $ζ$-regularized determinant of $L$ is defined by $\detz(L):=\exp(-ζ_L'(0)).$ 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 $\detz(L)=\frac{πW(ψ,ϕ)} {2^{ν_0+ν_1} Γ(ν_0+1)Γ(ν_1+1)}.$ Here $ϕ, ψ$ is a certain fundamental system of solutions for the homogeneous equation $Ly=0$, $W(ϕ, ψ)$ denotes their Wronski determinant, and $ν_0, ν_1$ are numbers related to the characteristic roots of the regular singular points $0, 1$.