On the spectrum of some Bloch-Torrey vector operators

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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