Bounds to unitary evolution

Upper and lower bounds are established for the survival probability $\mathrm{\mid }<\psi (0)\mathrm{\mid }\psi (t)>{\mathrm{\mid }}^2$ of a quantum state, in terms of the energy moments $<\psi (0)\mathrm{\mid }{H}^n\mathrm{\mid }\psi (0)>$. Introducing a cut-off in the energy generally enables considerable improvement in these bounds and allows the method to be used where the exact energy moments do not exist.