On the spectrum of some Bloch–Torrey vector operators

Abstract

We consider the Bloch–Torrey operator in $L^2(\mathrm{\Omega },{ℝ}^3)$, where $\mathrm{\Omega }\subseteq {ℝ}^k$. After normalization, this operator takes the form $-{𝜖}^2\mathrm{\Delta }+b\wedge$, where and $\mathit{b}$ represents a magnetic vector field. For $\mathrm{\Omega }={ℝ}^k$ we give natural conditions under which this operator can be defined as a maximally accretive operator, characterize its domain and obtain its spectral properties in some special cases where we manage to show that the essential spectrum is $[0,+\infty )$. This result lies in contrast with the $L^2(\mathrm{\Omega },{ℝ}^2)$ case considered in previous works.

In the asymptotic limit $𝜖\to 0$ and for $k=1$, assuming that $b(x)$ is an affine function, we give accurate estimates for the location of the discrete spectrum in the cases $\mathrm{\Omega }=ℝ$ or when $\mathrm{\Omega }$ is a finite interval. Resolvent estimates are established as well.