Help with this semiclassical sum please.

1. Feb 12, 2008

mhill

I know how to obtain a semiclassical expression for the sum:

$$\sum_{n=0}^{\infty} e^{-a E_{n}}\approx \iint _{\Gamma}e^{-aH(x,p)} dx, dp$$

where the integral is extended over the whole phase space of particle (x,p) , my question is if E-->oo is a big (but finite) energy , then what would be the asymptotic expression to evaluate:

$$\sum_{E_n \le E}e^{-aE_{n}}$$

this is just the sum of the function exp(-x) taken over all the Energies of Hamiltonian operator that are less or equal to a finite given Energy 'E' then what would be the expression for this last sum ?? , thanks in advance.

EDIT: i am not pretty sure, however i would say that if A is a constant then .. the last sum could be approximated by the integral:

$$\int_{0}^{N}exp(-E)dE$$ with E_n=E so 'N' is the N-th quantum

level whose energy is precisely 'E'

Last edited: Feb 12, 2008