The Method of Differential Inequalities

Abstract

In this paper we shall present a variant of the method of differential inequalities which, although much simpler than the original method of Mourre, allows one in the case of the Laplace operator (for example) to get the limiting absorption principle in the Besov space of Agmon and Hörmander and to obtain a large class of locally smooth operators. Both the local behaviour and the behaviour at infinity of the operators in this class seems to be optimal. We insist in this paper on the simplicity of the method, which explains why we consider a very restricted class of hamiltonians. In particular, the Mourre estimate is not explicitely needed. A much more general variant of the method of differential inequalities (representing a direct development of Mourre’s ideas), which enables us to obtain essentially optimal results in the N-body case, is presented in our paper Locally Smooth Operators and Limiting Absorption Principle for N-Body Hamiltonians (joint work with M.Mantoiu), which also contains references to other works. The method of the present note can be extended such as to give Hardy type inequalities for the Laplace operator in a very natural way (see our paper Mourre Estimates and Hardy Type Inequalities in collaboration with M.Mantoiu).