Is this licitthen why?(functional analysis question)

1. Sep 23, 2006

lokofer

I think i put this question before... but i can't be very sure.. let's suppose we have a Hamiltonian operator:

$$H= - \frac{d^2 }{dx^ 2}+V(x)$$ so its "energies" (eigenvalues)

satisfy that $$E(n)=-E(-n)$$ then here comes the question..is licit legal (at least as an approximation) to take:

$$Z(u)=\sum_{n=-\infty}^{n=\infty}e^{iuE(n)}\sim \iint dxdpe^{iup^2 +iuV(x)}$$ ??

The explanation is clear..you substitute a "discrete" sum over energies by a continous sum over all the energies..classically the energy of the system is $$E=p^2 +V(x)$$ (time independent potential) , so it would be similar to take the "sum-integral" approximation (valid at least at first order ??) i know that perhaps using functional analysis you could justify my approach, as a physicist when dealing with Statistical mechanics we use it all the time since "sums" are very hard to evaluate, except when E(n)=log(n) or E(n)=n :grumpy: :grumpy: usually the exponential is "real" but i think that for this case this wouldn't be a problem..also if we kenw Z(u) we could obtain the inverse of the potential taking:

$$A\int_{-\infty}^{\infty}du \frac{Z(u)e^{-iut}}{\sqrt (u) } = V^{-1}(t)$$ :tongue2: :tongue2:

Last edited: Sep 23, 2006
2. Sep 23, 2006

matt grime

Really, there is only a small number of times I can say "look up spectral analysis" before it starts becoming really really annoying, Jose.

3. Sep 23, 2006

lokofer

Sorry..i remember you that i'm not physicist..if you could recommend me a good "arxiv2 or other online paper about preliminaries of spectral analysis where my approximation is used and justified i would be very grateful... the only thing i know is that the "exponential sum" will be the trace of a certain operator $$Tr[e^{iu\hat H }$$

4. Sep 23, 2006

matt grime

Buy any text book on functional analysis, use google (spectral theory functional analysis), find someone's lecture notes. (and arxiv is not the p lace to learn about classical mathematics.) And this has nothing to do with you being, nor not being a physicist, but with you not listening to what people (shmoe, halls, hurkyl and me at anyrate) tell you.

5. Sep 23, 2006

lokofer

- I can't buy any book i'm actally unemployed

- perhaps a "trivial" solution to my problem is just put:

$$\sum_{n=-\infty}^{\infty}e^{iuE(n)} \sim \int_{E}dEe^{iuE}$$

if the "eigenvalue" are energies of a certain Hamiltonian operator $$H=E=p^2 + V(x)$$ then dE=dxdp (momentum and position) .

Using this is the only "justification", my teacher gave us when speaking about partition functions :tongue2: the problem here is that perhaps you can't use the Euler-Mc Laurin sum formula to improve the integral-series convergence.

6. Sep 23, 2006

Hurkyl

Staff Emeritus
Your school probably has a library...