Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Is this licitthen why?(functional analysis question)

  1. Sep 23, 2006 #1
    I think i put this question before...:confused: :confused: but i can't be very sure.. let's suppose we have a Hamiltonian operator:

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

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

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

    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 [tex] E=p^2 +V(x) [/tex] (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" :rolleyes: :rolleyes: 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:

    [tex] A\int_{-\infty}^{\infty}du \frac{Z(u)e^{-iut}}{\sqrt (u) } = V^{-1}(t) [/tex] :tongue2: :tongue2:
    Last edited: Sep 23, 2006
  2. jcsd
  3. Sep 23, 2006 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    Really, there is only a small number of times I can say "look up spectral analysis" before it starts becoming really really annoying, Jose.
  4. Sep 23, 2006 #3
    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...:rolleyes: :rolleyes: the only thing i know is that the "exponential sum" will be the trace of a certain operator [tex] Tr[e^{iu\hat H } [/tex]
  5. Sep 23, 2006 #4

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    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.
  6. Sep 23, 2006 #5
    - I can't buy any book i'm actally unemployed :frown: :frown:

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

    [tex] \sum_{n=-\infty}^{\infty}e^{iuE(n)} \sim \int_{E}dEe^{iuE} [/tex]

    if the "eigenvalue" are energies of a certain Hamiltonian operator [tex] H=E=p^2 + V(x) [/tex] 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.
  7. Sep 23, 2006 #6


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Your school probably has a library...
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook