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Is there a generalization to this ?

  1. Aug 18, 2007 #1
    Is there a generalization to this ??

    [tex] \sum_{n=-\infty}^{\infty} f(n)= \int_{-\infty}^{\infty}dx f(x) ( \sum_{m=-\infty}^{\infty} exp(2\pi i m) [/tex]

    if we consider that the expression [tex] \sum_{n}e^{2i \pi m} [/tex] is the trace (??) or whatever of a certain operator U (unitary) [tex] U=exp(iH) [/tex]

    so 'H' has an spectrum [tex] E_{n}=n [/tex] (Harmonic oscillator) then could it be generalized to include the 'Trace' of any H or even integral operator ??

    so it would read:

    [tex] \sum_{n=-\infty}^{\infty} f(n)= \int_{-\infty}^{\infty}dx f(x) \mathcal Tr[e^{2i\pi H}] [/tex]

    EDIT: = As a generalization i believe that

    [tex] \sum_{n=-\infty}^{\infty} f(E_{n})= \int_{-\infty}^{\infty}dx f(x) \mathcal Tr[e^{2ix\pi H}] [/tex]

    has this identity been discovered before ?? or i am the first to realize about this?, as always note that for the Harmonic Hamiltonian we recover usual Poisson sum formula
     
    Last edited: Aug 19, 2007
  2. jcsd
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