A new method of construction of resonances that applies to critical models

Abstract

We introduce a new method of multi-scale analysis that can be used to study the spectral properties of operators in non-relativistic quantum electrodynamics with critical coupling functions. We utilize our method to prove the existence of resonances of nonrelativistic atoms which are minimally coupled to the quantized (ultraviolet-regularized) radiation field and construct them together with the corresponding resonance eigenvector in case of critical coupling, i.e., without any infrared regularization. This result was first proved in [19] with the ingredient of a suitable Pauli-Fierz transformation. The purpose of the present paper, however, is to demonstrate the power of our new method for the estimation of resolvents that is based on the isospectral Feshbach-Schur map [8]. The reconstruction formula for the resolvent of an operator in terms of the resolvent of its image under the Feshbach-Schur map allows us to use a fixed projection and to compare two resolvents without actually decimating the degrees of freedom. This is in contrast to the renormalization group based on Feshbach-Schur map, developed in [8,5], that uses a decreasing sequence of ever-smaller projections and successively decimates the degrees of freedom. It is this new method that allows us to treat the critical and physically relevant Standard model of non-relativistic quantum electrodynamics[7] which is intractable by standard methods.