= Dirichlet oscillator determinant ratio
{c}
{title2=$\det(-\partial_t^2-\omega^2)/\det(-\partial_t^2)=\sin(\omega T)/(\omega T)$}
On $[0,T]$ with vanishing endpoint fluctuations, the sine-mode <eigenvalues> are $(\pi n/T)^2-\omega^2$. Divide by the free <eigenvalues> and use the <sine infinite product>. The ratio determines the <harmonic oscillator transition kernel> once the free <configuration-space path integral> is normalized. Zeros correspond to caustics where the fluctuation operator has a zero mode.
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