Dirichlet oscillator determinant ratio ($\det (-{\partial }_t^2-{\omega }^2)/\det (-{\partial }_t^2)=\sin (\omega T)/(\omega T)$)

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.