  We prove lower bounds on $||T_t||$, where $T_t$ is a one-parameter semigroup,
starting from information on the resolvent norms, i.e. the pseudospectra. We
provide a physically important example in which the growth of the semigroup
norm cannot be derived from spectral information. The generator is a
self-adjoint Schr\"odinger operator, but the semigroup growth is measured using
the $L^1$ norm rather than the $L^2$ norm. Numerical confirmation of the
results is provided.
