On the spectrum of some Bloch-Torrey vector operators

TL;DR

This paper analyzes the spectral properties of the Bloch-Torrey operator in different settings, revealing the presence of continuous and discrete spectra and characterizing the essential spectrum in higher dimensions.

Contribution

It extends previous spectral analysis of the Bloch-Torrey operator to the three-dimensional case and general setups, providing new insights into its spectrum and domain.

Findings

Real positive axis is in the continuous spectrum for the whole real line.

Discrete spectrum exists outside the real line.

The operator has an essential spectrum in general higher-dimensional settings.

Abstract

We consider the Bloch-Torrey operator in $L^2(I,{\mathbb R}^3)$ where $I\subseteq{\mathbb R}$. In contrast with the $L^2(I,{\mathbb R}^2)$ (as well as the $L^2({\mathbb R}^k,{\mathbb R}^2)$) case considered in previous works. We obtain that ${\mathbb R}_+$ is in the continuous spectrum for $I={\mathbb R}$ as well as discrete spectrum outside the real line. For a finite interval we find the left margin of the spectrum. In addition, we prove that the Bloch-Torrey operator must have an essential spectrum for a rather general setup in ${\mathbb R}^k$, and find an effective description for its domain.