Semiclassical exact expression ?

[zetafunction]

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

and let be a system so V(x)=V(-x) and V^{-1}(x)=\sqrt \pi \frac{d^{1/2}}{dx^{1/2}} N(x)<br /> <br /> then would it be true that the two function<br /> <br /> \sum_{n}exp(-tE_{n})=Z(t)= \int_{0}^{\infty}dN(x)exp(-tx)<br /> <br /> and \int_{0}^{\infty}dx\int_{0}^{\infty}dpexp(-tp^{2}-tV(x))=U(t)<br /> <br /> 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

 

[azntoon]

. Yes, they would be equal. To prove this, you would need to use the fact that for an even function, its Fourier transform is real and symmetric. This means that the Laplace transform of the even function can be written as a sum of two terms: one term containing the real part of the Fourier transform, and the other term containing the imaginary part. Using this, you can then show that the two functions are indeed equal.