Summary: 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 the 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 microscopic total Hamiltonian of an open quantum system reduces to the Jordan block form at an exceptional point in an exact manner that treats the continuum without any approximation by extending the problem to include eigenstates with complex eigenvalues that reside outside the Hilbert space. 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 total Hamiltonian. Because of the eigenvalue dependence of the effective Hamiltonian due to the dynamical nature of the coupling between the discrete states via the continuum states, a coalescence of eigenvalues results in the coalescence of the corresponding eigenvectors of the total Hamiltonian, which means that the system is at an exceptional point. 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.{
©2017 American Institute of Physics}