# Is there a generalization to this ?

1. Aug 18, 2007

### Sangoku

Is there a generalization to this ??

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

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

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

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

EDIT: = As a generalization i believe that

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

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