Spectral identities and smoothing estimates for evolution operators

TL;DR

This paper develops spectral identities and comparison principles to derive smoothing and decay estimates for evolution operators, with applications to fractional Laplacians, Stark Hamiltonians, and Schrödinger operators.

Contribution

It introduces spectral identities and comparison techniques for smoothing estimates, extending results to perturbed operators and various classes of evolution operators.

Findings

Derived spectral identities for evolution operators.

Established smoothing estimates for fractional Laplacians and Schrödinger operators.

Provided applications to perturbed operators and specific Hamiltonians.

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 our previous work), 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 Schr\"odinger operators with potentials.