Exponential Decay Laws in Perturbation Theory of Threshold and Embedded Eigenvalues

Abstract

Exponential decay laws for the metastable states resulting from perturbation of unstable eigenvalues are discussed. Eigenvalues embedded in the continuum as well as threshold eigenvalues are considered. Stationary methods are used, i.e. the evolution group is written in terms of the resolvent via Stone’s formula and a partition technique (Schur-Livsic-Feschbach-Grushin formula) is used to localize the essential terms. No analytic continuation of the resolvent is required. The main result is about the threshold case: for Schrodinger operators in odd dimensions the leading term of the life-time in the perturbation strength, e, is of order e 2+ν/2, where ν is an odd integer, ν≥−1. Examples covering all values of ν are given. For eigenvalues properly embedded in the continuum the results sharpen the previous ones.