The Fermi Golden Rule and its Form at Thresholds in Odd Dimensions

Abstract

Let H be a Schrodinger operator on a Hilbert space Open image in new window, such that zero is a nondegenerate threshold eigenvalue of H with eigenfunction Ψ0. Let W be a bounded selfadjoint operator satisfying 〈 Ψ0, WΨ0〈>0. Assume that the resolvent (H−z)−1 has an asymptotic expansion around z=0 of the form typical for Schrodinger operators on odd-dimensional spaces. Let H(ɛ) =H+ɛW for ɛ>0 and small. We show under some additional assumptions that the eigenvalue at zero becomes a resonance for H(ɛ), in the time-dependent sense introduced by A. Orth. No analytic continuation is needed. We show that the imaginary part of the resonance has a dependence on ɛ of the form ɛ2+(ν/2) with the integer ν≥−1 and odd. This shows how the Fermi Golden Rule has to be modified in the case of perturbation of a threshold eigenvalue. We give a number of explicit examples, where we compute the ``location'' of the resonance to leading order in ɛ. We also give results, in the case where the eigenvalue is embedded in the continuum, sharpening the existing ones.