Functional determinants via Wronski construction of Green functions

Abstract

A general technique is developed for calculating functional determinants of second-order differential operators with Dirichlet, periodic, and antiperiodic boundary conditions. As an example, we give simple formulas for a harmonic oscillator with an arbitrary time-dependent frequency. Here our result is a generalization of Gel'fand-Yaglom's famous formula which was restricted to Dirichlet boundary conditions. Apart from the generalization, our derivation is more transparent than theirs, the determinants requiring only knowledge of the classical trajectories. Special properties of operators with a zero mode are exhibited. Our technique does not require the calculation of the spectrum and is as simple as Wronski's method for Green functions.