Semigroup Growth Bounds

We prove lower bounds on $\mathrm{\mid }\mathrm{\mid }{T}_t\mathrm{\mid }\mathrm{\mid }$, 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.