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given a Hamiltonian and the semiclassical WKB partition function in units $ \hbar =2m=1 $

$ \Theta (t) = \frac{1}{2\pi} \iint dx dp exp(-tp^{2}-tV(x)) $

can i use this Theta function to evaluate the Funntional determinant ??

$ \zeta (s,z^{2}) \Gamma (s)= \int_{0}^{\infty}dt\Theta (t) exp(-tz^{2})$

$ \zeta (s,z^{2})= \sum_{n=0}^{\infty}(z^{2}+E_{n})^{-s}$

the functional determinant is evaluated as $ det(H+z^{2})=exp(- \partial _{s} \zeta (0,z^{2}))$

so we need only the Theta function which is $ \Theta (t)= \sum_{n}exp(-tE_{n})$ i have simply approximated this sum by an integral over phase space involving the Hamiltonian

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