  In this article we state and prove the spectral expansion of theta series
attached to the symmetric space $\mathrm{GL}_n(E)/\mathrm{GL}_n(F)$ where
$n\geq 1$ and $E/F$ is a quadratic extension of number fields. This is an
important step towards the fine spectral expansion of the Jacquet-Rallis trace
formula for general linear groups. To obtain our result, we extend the work of
Jacquet-Lapid-Rogawski on intertwining periods to the case of discrete
automorphic representations. The expansion we get is an absolutely convergent
integral of relative characters built upon Eisenstein series and intertwining
periods. We also establish a crucial but technical ingredient whose interest
lies beyond the focus of the article: we prove bounds for discrete Eisenstein
series on a neighborhood of the imaginary axis extending previous works of
Lapid on cuspidal Eisenstein series.
