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

We consider the Bloch–Torrey operator in L2(Ω, 3), where Ω k . After normalization, this operator takes the form 𝜖2Δ + b, where 𝜖 > 0 and b represents a magnetic vector field. For Ω = 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,+). This result lies in contrast with the L2(Ω, 2) case considered in previous works.

In the asymptotic limit 𝜖 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 Ω = or when Ω is a finite interval. Resolvent estimates are established as well.

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Bloch–Torrey, Schrödinger, matrix potential
Mathematical Subject Classification
Primary: 35P05
Received: 1 December 2020
Revised: 23 November 2021
Accepted: 5 January 2022
Published: 29 April 2022
Yaniv Almog
Department of Mathematics
Ort Braude College
Bernard Helffer
Laboratoire de Mathématiques Jean Leray
CNRS and Université de Nantes