The physical significance of exceptional points

Exceptional points are singularities that occur generically in the
spectrum and eigenfunctions of operators (usually matrices) that
depend on a parameter. For selfadjoint operators they always lie
in the complex plane of the parameter. Owing to their association
with level repulsion they feature prominently in quantum chaos.
Being singularities they play an essential role in a great variety
of approximation schemes. Recent experiments have nicely
illustrated the Riemann structure of the singularities (square
root type for the energies and fourth root type for the state
vectors) as well as the chiral character of the state vectors at
the singularities. The latter property depends crucially on the
particular type of the operator considered.