Almost Exponential Decay For A Coupled System Of Schrodinger Equations

Abstract

Let h = (h1 h2 ) be a resonant solution of a linearly coupled system of perturbed Schr odinger equations on the half line[0;1), with Dirichlet boundary conditions at the origin. The function h is then a generalized eigenvector of the related Hamil- tonian system H that satis es an outgoing condition at in nity. Then Hh = k2h, where the resonance k2 is complex with Im k2 negative. The main goal of this work is to show that the pres- ence of resonance is manifested by an approximate exponential behaviour. Indeed since the outgoing condition rules out square integrability, we truncate h to an interval containing the support of the perturbation and show that when the resonance is near the real axis, the probability amplitude ⟨h; e�����iHth⟩ has a certain ex- ponential behaviour in time.