Exceptional points and unitary evolution of the physical solutions

TL;DR

This paper explores exceptional points in a real pseudo-Hermitian Hamiltonian, revealing their impact on eigenfunctions, spectral properties, and the unitary or pseudounitary evolution of scattering states.

Contribution

It presents the first example of exceptional points in the continuous spectrum of a real, pseudo-Hermitian Hamiltonian and analyzes their effects on eigenfunctions and time evolution.

Findings

Exceptional points are linked to double poles in the normalization factor.

At exceptional points, eigenfunctions coalesce into Jordan cycles.

Regular scattering eigenfunctions evolve unitarily, irregular ones pseudounitarily.

Abstract

An example of exceptional points in the continuous spectrum of a real, pseudo-Hermitian Hamiltonian of von Neumann-Wigner type is presented and discussed. Remarkably, these exceptional points are associated with a double pole in the normalization factor of the Jost eigenfunctions normalized to unit flux at infinity. At the exceptional points, the two unnormalized Jost eigenfunctions are no longer linearly independent but coalesce to give rise to two Jordan cycles of generalized bound state eigenfunctions embedded in the continuum and a Jordan block representation of the Hamiltonian. The regular scattering eigenfunction vanishes at the exceptional point and the irregular scattering eigenfunction has a double pole at that point. In consequence, the time evolution of the regular scattering eigenfunction is unitary, while the time evolution of the irregular scattering eigenfunction is pseudounitary. The scattering matrix is a regular analytical function of the wave number $k$ for all $k$ including the exceptional points.