Semiclassical exact expression ?

[zetafunction]

let be $$N(x)= \sum_{n} H(x-E_{n})$$ the eingenvalue 'staircase' function

and let be a system so [tex] V(x)=V(-x) [tex]and $$V^{-1}(x)=\sqrt \pi \frac{d^{1/2}}{dx^{1/2}} N(x)$$

then would it be true that the two function

$$\sum_{n}exp(-tE_{n})=Z(t)= \int_{0}^{\infty}dN(x)exp(-tx)$$

and $$\int_{0}^{\infty}dx\int_{0}^{\infty}dpexp(-tp^{2}-tV(x))=U(t)$$

would be equal ?? i have just compared the two results $$\int_{0}^{\infty}dnN(x)exp(-tx)=\int_{0}^{\infty}dx\int_{0}^{\infty}dpexp(-tp^{2}-tV(x))$$ i have taken the Laplace transform inside and get the desired result assuming that the potential V(x) is EVEN