Spectral branch points of the Bloch-Torrey operator

TL;DR

This paper explores spectral branch points in non-Hermitian operators, especially the Bloch-Torrey operator, revealing their mathematical properties, physical implications, and providing an efficient numerical method to identify these points relevant for magnetic resonance.

Contribution

It introduces a numerical algorithm to locate spectral branch points and applies it to the Bloch-Torrey operator, linking spectral features to physical phenomena in magnetic resonance.

Findings

Spectral branch points cause non-analyticities affecting eigenvalue expansions.

The algorithm efficiently finds branch points in complex spectra.

Spectral branch points influence diffusion magnetic resonance experiments.

Abstract

We investigate the peculiar feature of non-Hermitian operators, namely, the existence of spectral branch points (also known as exceptional or level crossing points), at which two (or many) eigenmodes collapse onto a single eigenmode and thus loose their completeness. Such branch points are generic and produce non-analyticities in the spectrum of the operator, which, in turn, result in a finite convergence radius of perturbative expansions based on eigenvalues and eigenmodes that can be relevant even for Hermitian operators. We start with a pedagogic introduction to this phenomenon by considering the case of $2\times 2$ matrices and explaining how the analysis of more general differential operators can be reduced to this setting. We propose an efficient numerical algorithm to find spectral branch points in the complex plane. This algorithm is then employed to show the emergence of spectral branch points in the spectrum of the Bloch-Torrey operator $-\nabla^2 - igx$, which governs the time evolution of the nuclear magnetization under diffusion and precession. We discuss their mathematical properties and physical implications for diffusion nuclear magnetic resonance experiments in general bounded domains.