Beyond semiclassical approach

zetafunction

let be [tex] E_n [/tex] the set of values of a given Hamiltonian H=H(p,q) , then if we wished to calculate the following sum

[tex] \sum_{n >0} exp (-uE_n ) [/tex] for real valued and positive 'u' then the semiclassical approximation gives

[tex] \sum_{n >0} exp (-uE_n ) = \iint _{P}dqdp exp(-uH(q,p)) [/tex]

is this the best approximation we can use ?,