Functional determinants from Wronski Green functions

Abstract

A general technique is developed for calculating functional determinants of second-order differential operators with Dirichlet, periodic, and antiperiodic boundary conditions, without the knowledge of spectral properties. As an example, we give explicit formulas for a harmonic oscillator with an arbitrary time-dependent frequency, where our result is a generalization of the Gel’fand–Yaglom famous formula for Dirichlet boundary conditions. Our technique is based on the Wronski’s construction of Green functions, which does not require spectral knowledge. Our final formula expresses the ratios of functional determinants in terms of an ordinary 2×2 determinant of a constant matrix constructed from two linearly independent solutions of the homogeneous differential equations associated with second-order differential operators. For ratios of determinants encountered in semiclassical fluctuations around a classical solution, the result can further be expressed in terms of the classical solution. Special properties of operators with a zero mode are exhibited.