  Consider the Langevin process, described by a vector (position,momentum) in
$\mathbb{R}^{d}\times\mathbb{R}^d$. Let $\mathcal O$ be a $\mathcal{C}^2$ open
bounded and connected set of $\mathbb{R}^d$. We prove the compactness of the
semigroup of the Langevin process absorbed at the boundary of the domain
$D:=\mathcal{O}\times\mathbb{R}^d$. We then obtain the existence of a unique
quasi-stationary distribution (QSD) for the Langevin process on $D$. We also
provide a spectral interpretation of this QSD and obtain an exponential
convergence of the Langevin process conditioned on non-absorption towards the
QSD.
