| The time evolution of a Schrodinger operator H representing the energy of a particle in an initial state {dollar}varphi{dollar} is {dollar}esp{lcub}-iHt{rcub}varphi{dollar} and the probability that the particle remains in the state at time t is {dollar}vert vertsp2{dollar}. When a potential has compact support and H has a resonance, this probability exponentially decays with error {dollar}O(vertepsilonlog epsilonvert){dollar} where {dollar}epsilon{dollar} is the imaginary part of a resonance and {dollar}varphi{dollar} is the cut-off resonance eigenfunction. This idea is extended to show a similar result for a potential which goes to zero at infinity and is convex outside an interval. In this generality resonance cannot be defined. Also the existence of outgoing solutions and a solution with approximately exponential time decay for such a potential is shown. |
