Maximal couplings in PT-symmetric chain-models with the real spectrum of energies

TL;DR

This paper investigates the parameter domain ensuring real energy spectra in PT-symmetric chain-model Hamiltonians, identifying extremal coupling points where the boundary of this domain contacts geometric shapes.

Contribution

It characterizes the boundary of the coupling parameter domain for PT-symmetric Hamiltonians with real spectra across all matrix sizes, identifying extremal points geometrically.

Findings

Identified extremal points of the coupling domain boundary.

Described boundary contact points with geometric shapes.

Analyzed domain structure for all matrix sizes.

Abstract

The domain ${\cal D}$ of all the coupling strengths compatible with the reality of the energies is studied for a family of non-Hermitian $N$ by $N$ matrix Hamiltonians $H^{(N)}$ with tridiagonal and ${\cal PT}-$symmetric structure. At all dimensions $N$, the coordinates are found of the extremal points at which the boundary hypersurface $\partial {\cal D}$ touches the circumscribed sphere (for odd $N=2M+1$) or ellipsoid (for even $N=2K$).