Nonexponential decay laws in perturbation theory of near threshold eigenvalues

Abstract

We consider a two channel model of the form Hε=[Hop00E0]+ε[0W12W210] on H=Hop⊕C. The operator Hop is assumed to have the properties of a Schrödinger operator in odd dimensions, with a threshold at zero. As the energy parameter E0 is tuned past the threshold, we consider the survival probability |⟨Ψ0,e−itHεΨ0⟩|2, where Ψ0 is the eigenfunction corresponding to eigenvalue E0 for ε=0. We find nonexponential decay laws for ε small and E0 close to zero provided that the resolvent of Hop is not at least Lipschitz continuous at the threshold zero.