Unitarity corridors to exceptional points

2019/09/04 by Miloslav Znojil
Mathematics · Physics and Astronomy · #Hamiltonian (control theory) #Hermitian matrix #Hilbert space #Lambda #Mathematical physics #Mathematics #Nonlinear Photonic Systems #Physics #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum mechanics #Space (punctuation) #Unitarity #quant-ph

paper · pdf · doi:10.1103/physreva.100.032124

published as Physical Review A 100, 032124 {2019) · 21 pp., 1 fig

arxiv created 2019/09/04 · openalex publication_date 2019/09/26 · arxiv updated 2019/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Phenomenological quantum Hamiltonians H^(N)(\ensuremathλ)=J^(N)+\ensuremathλ\phantom\rule0.16em0exV^(N)(\ensuremathλ) representing a general real N²-parametric perturbation of an exceptional-point-related unperturbed Jordan-block Hamiltonian J^(N) are considered. Tractable as non-Hermitian (in a preselected, unphysical Hilbert space) as well as, simultaneously, Hermitian (in another, ``physical'' Hilbert space), these matrices may represent a unitary, closed quantum system if and only if the spectrum is real. At small \ensuremathλ we show that the parameters are then confined to a ``stability corridor'' S of the access to the extreme dynamical exceptional-point \ensuremathλ\ensuremath→0 regime. The corridors are narrow and N-dependent: they are formed by multiscale perturbations which are small in physical Hilbert space, i.e., which are such that \ensuremathλ\phantom\rule0.16em0exV_j+k,j^(N)(\ensuremathλ)=O(\ensuremathλ^(k+1)/2)\phantom\rule0.16em0ex at k=1,2,...,N\ensuremath-1\phantom\rule0.16em0ex and all j.

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