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.
Categories: math-ph math.MP