Zeroth order conjugate operator in N-body Schrödinger operators

Kenichi Ito, University of Tokyo, Japan

Abstract

We develop a scheme of proofs for spectral theory of the N-body Schrödinger operators, reproducing and extending a series of sharp results under minimum conditions. The main results are Rellich's theorem and the limiting absorption principle. We present a new proof of Rellich's theorem which is unified with exponential decay estimates studied previously only for L²-eigenfunctions. Each pair-potential is a sum of a long-range term with first order derivatives, a short-range term without derivatives and a singular term of operator- or form-bounded type. The setup can also include hard-core interactions. Our proof consists of a systematic use of commutators with `zeroth order' operator, not like the standard `first order' conjugate operator in the Mourre theory. In particular, our proofs do not rely on Mourre's differential inequality technique. This talk is based on a recent joint work with T. Adachi, K. Itakura and E. Skibsted.