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arXiv · 2312.04200

Spectral properties of the Bloch-Torrey operator in three dimensions

Denis S. Grebenkov
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Abstract

We consider the Bloch-Torrey operator, $-\Delta + igx$, 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.

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Denis S. Grebenkov. 2023-12-07. Spectral properties of the Bloch-Torrey operator in three dimensions. https://doi.org/10.1088/1751-8121%2Fad2d6d

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