Tridiagonal PT-symmetric N by N Hamiltonians and a fine-tuning of their observability domains in the strongly non-Hermitian regime

TL;DR

This paper investigates N-dimensional PT-symmetric tridiagonal Hamiltonians, revealing how specific coupling alignments in the strongly non-Hermitian regime shape the domain where the spectrum remains real and observable.

Contribution

It introduces a method to analyze the spectral reality domain of PT-symmetric Hamiltonians with special coupling configurations in the non-Hermitian regime.

Findings

Secular equation factorizes in the strongly non-Hermitian regime.

Fine-tuned coupling alignment leads to sharply spiked boundary of observability domain.

Asymptotic shape of the quasi-Hermiticity domain boundary is characterized.

Abstract

A generic PT-symmetric Hamiltonian is assumed tridiagonalized and truncated to N dimensions, and its up-down symmetrized special cases with J=[N/2] real couplings are considered. In the strongly non-Hermitian regime the secular equation gets partially factorized at all N. This enables us to reveal a fine-tuned alignment of the dominant couplings implying an asymptotically sharply spiked shape of the boundary of the J-dimensional quasi-Hermiticity domain in which all the spectrum of energies remains real and observable.