Spectral properties of the Bloch–Torrey operator in three dimensions

We consider the Bloch–Torrey operator, − Δ + i g x , that governs the time evolution of the transverse magnetization in diffusion magnetic resonance imaging (dMRI). Using the matrix formalism, we compute numerically the eigenvalues and eigenfunctions of this non-Hermitian operator for two bounded th...

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Bibliographic Details
Published in:Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2024-03, Vol.57 (12), p.125201
Main Author: Grebenkov, Denis S
Format: Article
Language:English
Subjects:
Bloch–Torrey operator
branch point
diffusion-weighted NMR
localization
microstructure
non-Hermitian operator
Physics
pulsed-gradient spin-echo
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Summary:We consider the Bloch–Torrey operator, − Δ + i g x , that governs the time evolution of the transverse magnetization in diffusion magnetic resonance imaging (dMRI). Using the matrix formalism, we compute numerically the eigenvalues and eigenfunctions of this non-Hermitian operator for two bounded three-dimensional domains: a sphere and a capped cylinder. We study the dependence of its eigenvalues and eigenfunctions on the parameter g and on the shape of the domain (its eventual symmetries and anisotropy). In particular, we show how an eigenfunction drastically changes its shape when the associated eigenvalue crosses a branch (or exceptional) point in the spectrum. Potential implications of this behavior for dMRI are discussed.
ISSN:1751-8113
1751-8121
DOI:10.1088/1751-8121/ad2d6d