#### Vol. 4, No. 1, 2022

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On the spectrum of some Bloch–Torrey vector operators

### Yaniv Almog and Bernard Helffer

Vol. 4 (2022), No. 1, 1–48
##### Abstract

We consider the Bloch–Torrey operator in ${L}^{2}\left(\mathrm{\Omega },{ℝ}^{3}\right)$, where $\mathrm{\Omega }\subseteq {ℝ}^{k}$. After normalization, this operator takes the form $-{𝜖}^{2}\mathrm{\Delta }+b\wedge$, where $𝜖>0$ and $b$ 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 $\left[0,+\infty \right)$. This result lies in contrast with the ${L}^{2}\left(\mathrm{\Omega },{ℝ}^{2}\right)$ case considered in previous works.

In the asymptotic limit $𝜖\to 0$ and for $k=1$, assuming that $b\left(x\right)$ 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.

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