# Exact description of coalescing eigenstates in open quantum systems in terms of microscopic Hamiltonian dynamics

## TL;DR

This paper presents an exact method to describe coalescing eigenstates at exceptional points in open quantum systems using microscopic Hamiltonian dynamics, avoiding phenomenological approximations and ensuring continuity of observable quantities.

## Contribution

It introduces a rigorous approach to derive Jordan block form at exceptional points directly from the microscopic Hamiltonian, including continuum states, without approximations.

## Key findings

- Exact Jordan form derivation at exceptional points
- Extension of eigenstates outside Hilbert space
- Continuity of observables across exceptional points

## 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.