  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.
