  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.
