semiclassical exact expression (in one dimension only)

Question

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) $

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

Answer 1

The answer is in general No, for various reasons.

1. The WKB approximation is generally not exact for a finite energy-level $N$.

2. Point 1 is true even if we include the metaplectic correction from the Maslov index.

3. OP's sought-for formula is not exact for a finite energy-level $N$ even for the simple quantum harmonic oscillator (SHO). (This is despite the fact that the WKB approximation with the metaplectic correction famously yields the exact SHO spectrum.) To derive the correct parabolic potential via the fractional derivative formula$^1$, one should use an $N(E)$ function that is affine in the energy $E$, not the staircase function $N(E)=\sum_n \Theta(E-E_n)$. (Of course in the harmonic oscillator case, the staircase function can be approximated by an affine function in the continuum limit.)

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$^1$See e.g. this answer for more details.