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arXiv · 2009.03036

On the spectrum of some Bloch-Torrey vector operators

Yaniv Almog·Bernard Helffer
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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.

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Yaniv Almog, Bernard Helffer. 2020-09-07. On the spectrum of some Bloch-Torrey vector operators. https://doi.org/10.2140/paa.2022.4.1

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