Determinants of regular singular Sturm-Liouville operators

We consider a regular singular Sturm-Liouville operator $L:=-\frac{d^2}{dx^2}

\frac{q(x)}{x^2 (1-x)^2} $o n t h e l i n e s e g m e n t$ [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 $ζ_{L} (s) = \sum_{λ \in \spec (L) ∖ {0}} λ^{- s}$ 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 $L y = 0$ 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)}. $H e r e$ \phi, \psi$ is a certain fundamental system of solutions for the homogeneous equation $L y = 0$ , $W (ϕ, ψ)$ denotes their Wronski determinant, and $ν_{0}, ν_{1}$ are numbers related to the characteristic roots of the regular singular points $0, 1$.

We consider a regular singular Sturm-Liouville operator $L:=-\frac{d^2}{dx^2}

\frac{q(x)}{x^2 (1-x)^2} $o n t h e l i n e s e g m e n t$ [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 $ζ_{L} (s) = \sum_{λ \in \spec (L) ∖ {0}} λ^{- s}$ 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 $L y = 0$ 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)}. $H e r e$ \phi, \psi$ is a certain fundamental system of solutions for the homogeneous equation $L y = 0$ , $W (ϕ, ψ)$ denotes their Wronski determinant, and $ν_{0}, ν_{1}$ are numbers related to the characteristic roots of the regular singular points $0, 1$.

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