Poisson summation formula

In this note we will follow the presentation given by Ziman .

Consider \(f(x)\) an arbitrary function defined in the interval \([n,n+1]\). We can write

\begin{equation}f(x)=\sum_{s=-\infty}^{+\infty}e^{-i2\pi xs}g_s\label{fourier}\end{equation}

with \(g_s\) given by

\begin{equation} g_s=\int_n^{n+1} dx f(x) e^{i2\pi sx} \label{fourier-coefficient}\;.\end{equation}

We can evaluate Eq. \eqref{fourier} at the point \(n+\gamma\) with \(\gamma\in[0,1]\) to obtain

\begin{equation}f(n+\gamma)=\sum_{s=-\infty}^{+\infty} e^{i2\pi s\gamma} \int_n^{n+1} dx f(x) e^{i2\pi sx}\;.\label{fn}\end{equation}

Now we can sum over \(n\) to obtain the Poisson summation formula

\begin{equation}\sum_{n=-\infty}^{+\infty}f(n+\gamma)=\sum_{s=-\infty}^{+\infty} e^{i2\pi s\gamma} \int_{-\infty}^{+\infty} dx f(x) e^{i2\pi sx}\;.\label{pois}\end{equation}

Application: A very useful identity in Condensed Matter Physics

If \(f(x)=\delta(x)\) then
\begin{equation}\sum_{n=-\infty}^{+\infty}\delta(n+\gamma)=\sum_{s=-\infty}^{+\infty} e^{i2\pi s\gamma} \;.\label{conmat}\end{equation}

Application: Jacobi imaginary transformation

We can use the Poisson summation formula in Eq. \eqref{pois} just derived to obtain the formula used in the evaluation of Ewald sums.

\[\sum_{k=-\infty}^{+\infty} e^{-k^2t}=\sqrt{\pi \over t}\sum_{p=-\infty}^{+\infty} e^{-{\pi^2 p^2\over t}}\label{jacobi}\]


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