Spectral branch points of the Bloch–Torrey operator

We investigate the peculiar feature of non-Hermitian operators, namely, existence 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 are generic produce non-analyticities in spectrum operator, which, turn, result finite convergence radius perturbative expansions based on eigenvalues that can be relevant even for Hermitian operators. start with pedagogic introduction to this phenomenon by considering case $2\times 2$ matrices explaining how analysis more general differential operators reduced setting. propose an efficient numerical algorithm find complex plane. This is then employed show emergence Bloch-Torrey operator $-\nabla^2 - igx$, governs time evolution nuclear magnetization under diffusion precession. discuss mathematical properties physical implications magnetic resonance experiments bounded domains.