let be [tex] N(x)= \sum_{n} H(x-E_{n}) [/tex] the eingenvalue 'staircase' function(adsbygoogle = window.adsbygoogle || []).push({});

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

then would it be true that the two function

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

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

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

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# Semiclassical exact expression ?

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