  We study the behaviour of the norm of the resolvent for non-self-adjoint
operators of the form $A := -\partial_x + W(x)$, with $W(x) \ge 0$, defined in
$L^2(\mathbb{R})$. We provide a sharp estimate for the norm of its resolvent
operator, $\| (A - \lambda)^{-1} \|$, as the spectral parameter diverges
$(\lambda \to +\infty)$. Furthermore, we describe the $C_0$-semigroup generated
by $-A$ and determine its norm. Finally, we discuss the applications of the
results to the asymptotic description of pseudospectra of Schr\"odinger and
damped wave operators and also the optimality of abstract resolvent bounds
based on Carleman-type estimates.
