We consider the Bloch–Torrey operator, −Δ+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.

中文翻译：

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三维布洛赫-托里算子的谱特性

我们考虑布洛赫-托里算子， -Δ+我GX ，它控制扩散磁共振成像 (dMRI) 中横向磁化强度的时间演化。使用矩阵形式，我们以数值方式计算两个有界三维域（球体和有盖圆柱体）的非厄米算子的特征值和特征函数。我们研究其特征值和特征函数对参数的依赖性G以及域的形状（其最终的对称性和各向异性）。特别是，我们展示了当相关特征值穿过频谱中的分支（或例外）点时，特征函数如何急剧改变其形状。讨论了这种行为对 dMRI 的潜在影响。