Eigenfunctions decay for magnetic pseudodifferential operators

TL;DR

This paper proves that eigenfunctions of certain magnetic pseudodifferential operators, including the relativistic Schrödinger operator, decay rapidly or exponentially, enhancing understanding of their spectral properties.

Contribution

It establishes decay rates of eigenfunctions for a broad class of magnetic pseudodifferential operators, including the relativistic Schrödinger operator with magnetic fields.

Findings

Eigenfunctions decay rapidly for the studied operators.

Exponential decay is achieved under stronger assumptions.

Results apply to a class of operators including the relativistic Schrödinger operator.

Abstract

We prove rapid decay (even exponential decay under some stronger assumptions) of the eigenfunctions associated to discrete eigenvalues, for a class of self-adjoint operators in $L^2(\mathbb{R}^d)$ defined by ``magnetic'' pseudodifferential operators (studied in \cite{IMP1}). This class contains the relativistic Schr\"{o}dinger operator with magnetic field.