1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?
Draft saved Draft deleted



Similar Discussions: Is there a generalization to this ?
  1. General Form (Replies: 3)

Loading...