Title: Exact description of coalescing eigenstates in open quantum systems in terms of microscopic Hamiltonian dynamics,
Abstract: At the exceptional point where two eigenstates coalesce in open quantum
systems, the usual diagonalization scheme breaks down and the Hamiltonian can
only be reduced to Jordan block form. Most of the studies on the exceptional
point appearing in the literature introduce a phenomenological effective
Hamiltonian that essentially reduces the problem to that of a finite
non-Hermitian matrix for which it is straightforward to obtain the Jordan form.
In this paper, we demonstrate how the Hamiltonian of an open quantum system
reduces to Jordan block form at an exceptional point in an exact manner that
treats the continuum without any approximation. Our method relies on the
Brillouin-Wigner-Feshbach projection method according to which we can obtain a
finite dimensional effective Hamiltonian that shares the discrete sector of the
spectrum with the original Hamiltonian. While owing to its eigenvalue
dependence this effective Hamiltonian cannot be used to write the Jordan block
directly, we show that by formally extending the problem to include eigenstates
with complex eigenvalues that reside outside the usual Hilbert space, we can
obtain the Jordan block form at the exceptional point without introducing any
approximation. We also introduce an extended Jordan form basis away from the
exceptional point, which provides an alternative way to obtain the Jordan block
at an exceptional point. The extended Jordan block connects continuously to the
Jordan block exactly at the exceptional point implying that the observable
quantities are continuous at the exceptional point.