On the Essential Spectrum of Phase-Space Anisotropic Pseudodifferential Operators

TL;DR

This paper characterizes the essential spectrum of phase-space anisotropic pseudodifferential operators in L^2(R^n) by analyzing their asymptotic localizations, extending previous results from purely configurational anisotropic operators.

Contribution

It extends the analysis of anisotropic operators to include phase-space behavior, providing a new description of their essential spectrum via ultrafilter limits.

Findings

The essential spectrum is the closure of the union of spectra of phase-space localizations.

The approach generalizes previous results limited to spatial infinity behavior.

Provides a framework for understanding phase-space anisotropic operators in quantum mechanics.

Abstract

A phase-space anisotropic operator in H=L^2(R^n) is a self-adjoint operator whose resolvent family belongs to a natural C*-completion of the space of H\"ormander symbols of order zero. Equivalently, each member of the resolvent family is norm-continuous under conjugation with the Schr\"odinger unitary representation of the Heisenberg group. The essential spectrum of such a phase-space anisotropic operator is the closure of the union of usual spectra of all its "phase-space asymptotic localizations", obtained as limits over diverging ultrafilters of R^{n}\times R^n-translations of the operator. The result extends previous analysis of the purely configurational anisotropic operators,for which only the behavior at infinity in R^n was allowed to bo non-trivial.