  Let $\pi$ be a positive continuous target density on $\mathbb{R}$. Let $P$ be
the Metropolis-Hastings operator on the Lebesgue space $\mathbb{L}^2(\pi)$
corresponding to a proposal Markov kernel $Q$ on $\mathbb{R}$. When using the
quasi-compactness method to estimate the spectral gap of $P$, a mandatory first
step is to obtain an accurate bound of the essential spectral radius
$r\_{ess}(P)$ of $P$. In this paper a computable bound of $r\_{ess}(P)$ is
obtained under the following assumption on the proposal kernel: $Q$ has a
bounded continuous density $q(x,y)$ on $\mathbb{R}^2$ satisfying the following
finite range assumption : $|u| \textgreater{} s \, \Rightarrow\, q(x,x+u) = 0$
(for some $s\textgreater{}0$). This result is illustrated with Random Walk
Metropolis-Hastings kernels.
