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Supersymmetry and exceptional points

Miloslav Znojil

TL;DR

The paper addresses how energy degeneracies attributed to SUSY and those arising at Kato exceptional points (EPs) can be unified within the three-Hilbert-space ($\mathcal{K}$, $\mathcal{L}$, $\mathcal{H}$) / quasi-Hermitian framework. It develops this bridge by detailing hidden Hermiticity, the Dyson map, and a metric $\Theta=\Omega^{\dagger}\Omega$ that preserves unitarity in a physically meaningful space, then analyzes EPs via simple two-by-two models and real-spectrum domains in crypto-Hermitian local-potentials. A key contribution is the construction of infinite families of SUSY-EP separated, regularized spiked harmonic oscillators, with exact spectra $E_{Q,n}=2-2Q\alpha+4n$ (where $\alpha=\ell+1/2$ and $Q=\pm1$) and explicit solvable cases that illustrate how SUSY and EP physics co-mingle. This work broadens the scope of SUSY in non-Hermitian contexts and clarifies how unitarity and observability are constrained by EP boundaries, offering tractable models for exploring these foundational ideas.

Abstract

A conceptual bridge is provided between SUSY and the three-Hilbert-space upgrade of quantum theory a.k.a. ${\cal PT}-$symmetric or quasi-Hermitian. In particular, a natural theoretical link is found between SUSY and the presence of Kato's exceptional points (EPs), both being related to the phenomenon of degeneracy of energy levels. Regularized spiked harmonic oscillator is recalled for illustration.

Supersymmetry and exceptional points

TL;DR

The paper addresses how energy degeneracies attributed to SUSY and those arising at Kato exceptional points (EPs) can be unified within the three-Hilbert-space (, , ) / quasi-Hermitian framework. It develops this bridge by detailing hidden Hermiticity, the Dyson map, and a metric that preserves unitarity in a physically meaningful space, then analyzes EPs via simple two-by-two models and real-spectrum domains in crypto-Hermitian local-potentials. A key contribution is the construction of infinite families of SUSY-EP separated, regularized spiked harmonic oscillators, with exact spectra (where and ) and explicit solvable cases that illustrate how SUSY and EP physics co-mingle. This work broadens the scope of SUSY in non-Hermitian contexts and clarifies how unitarity and observability are constrained by EP boundaries, offering tractable models for exploring these foundational ideas.

Abstract

A conceptual bridge is provided between SUSY and the three-Hilbert-space upgrade of quantum theory a.k.a. symmetric or quasi-Hermitian. In particular, a natural theoretical link is found between SUSY and the presence of Kato's exceptional points (EPs), both being related to the phenomenon of degeneracy of energy levels. Regularized spiked harmonic oscillator is recalled for illustration.

Paper Structure

This paper contains 10 sections, 2 theorems, 17 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The concept of hiddenly Hermitian Hamiltonians
  3. Non-Hermitian physics: closed versus open systems
  4. Triplets of Hilbert spaces: A key to the unitarity paradox
  5. Non-uniqueness of Hilbert space ${\cal H}$
  6. The concept of exceptional points
  7. The birth of the concept in perturbation theory
  8. EPs in two by two effective Hamiltonians
  9. EPs in crypto-Hermitian models with local potentials
  10. Regularized harmonic oscillator

Key Result

Lemma 1

[a2] The observable predictions concerning a closed quantum system and using the space-Hamiltonian doublets $({\cal H},H)$ or $({\cal L},\mathfrak{h})$ are equivalent.

Figures (1)

  • Figure :

Theorems & Definitions (2)

  • Lemma 1
  • Lemma 2