Quantum catastrophes: a case study

TL;DR

This paper investigates quantum catastrophes through the lens of crypto-Hermitian Hamiltonians, analyzing how bound-state spectra degenerate and complexify at exceptional points using effective matrix models and constructing unique metrics.

Contribution

It introduces a framework for studying quantum catastrophes via N-by-N matrix Hamiltonians and develops a method to construct unique, extrapolation-friendly metrics for these systems.

Findings

Identification of N-plet exceptional point crossings leading to spectral degeneracies.

Development of a closed-form, unique metric for the Hamiltonian.

Simulation of spectral degenerations using benchmark matrix models.

Abstract

The bound-state spectrum of a Hamiltonian H is assumed real in a non-empty domain D of physical values of parameters. This means that for these parameters, H may be called crypto-Hermitian, i.e., made Hermitian via an {\it ad hoc} choice of the inner product in the physical Hilbert space of quantum bound states (i.e., via an {\it ad hoc} construction of the so called metric). The name of quantum catastrophe is then assigned to the N-tuple-exceptional-point crossing, i.e., to the scenario in which we leave domain D along such a path that at the boundary of D, an N-plet of bound state energies degenerates and, subsequently, complexifies. At any fixed $N \geq 2$, this process is simulated via an N by N benchmark effective matrix Hamiltonian H. Finally, it is being assigned such a closed-form metric which is made unique via an N-extrapolation-friendliness requirement.