We present main ideas in the spectral and pseudospectral analysis of the damped wave equation $u_{tt} + 2a(x) u_t = \Delta_x u$ with an unbounded damping coefficient $a$, e.g.~$a(x) = x^2$, $x \in \mathbb R$. The key step is the study of the associated quadratic operator function $T(\lambda) = -\Delta + 2 \lambda a(x) + \lambda^2$, $\lambda \in \mathbb C$. Due to the form of this function, some of the recently developed techniques of the spectral theory for Schr\"odinger operator with complex potential can be generalized; in particular the pseudomode construction and upper resolvent norm estimates. Moreover, the latter and recent results in semigroup theory provide an estimate on the decay of solutions as $t \to + \infty$.
The talk is based mainly on joint works with A. Arnal.