Spectral identities and smoothing estimates for evolution operators

Abstract

Smoothing (and decay) spacetime estimates are discussed for evolution groups of self-adjoint operators in an abstract setting. The basic assumption is the existence (and weak continuity) of the spectral density in a functional setting. Spectral identities for the time evolution of such operators are derived, enabling results concerning "best constants" for smoothing estimates. When combined with suitable "comparison principles" (analogous to those established in [22]), they yield smoothing estimates for classes of functions of the operators .A important particular case is the derivation of global spacetime estimates for a perturbed operator H + V on the basis of its comparison with the unperturbed operator H.A number of applications are given, including smoothing estimates for fractional Laplacians, Stark Hamiltonians and Schrödinger operators with potentials. Contents1. Introduction 2 2. The basic setup and notation 3 3. Global smoothing estimates 4 4. Comparison principles 9 4.1. Comparing unperturbed and perturbed operators 11 5. Applications to operators of mathematical physics 13 5.1. The fractional Laplacian 13 5.2. The Stark Hamiltonian 13 5.3. The Schrödinger operator with potential 14 References 15