# PT-symmetry entails pseudo-Hermiticity regardless of diagonalizability

## TL;DR

This paper proves that in finite-dimensional systems, PT-symmetric Hamiltonians are inherently pseudo-Hermitian regardless of diagonalizability, linking PT-symmetry breaking to system instabilities and eigenmode resonances.

## Contribution

It establishes that PT-symmetry implies pseudo-Hermiticity without requiring diagonalizability, extending previous results and connecting symmetry breaking to instability mechanisms.

## Key findings

- PT-symmetric Hamiltonians are pseudo-Hermitian regardless of diagonalizability
- PT-symmetry breaking corresponds to instabilities in pseudo-Hermitian systems
- PT-symmetry breaking involves resonance between eigenmodes with different Krein signatures

## Abstract

We prove that in finite dimensions, a Parity-Time (PT)-symmetric Hamiltonian is necessarily pseudo-Hermitian regardless of whether it is diagonalizable or not. This result is different from Mostafazadeh's, which requires the Hamiltonian to be diagonalizable. PT-symmetry breaking often occurs at exceptional points where the Hamiltonian is not diagonalizable. Our result implies that PT-symmetry breaking is equivalent to the onset of instabilities of pseudo-Hermitian systems, which was systematically studied by Krein et al. in 1950s. In particular, we show that the mechanism of PT-symmetry breaking is the resonance between eigenmodes with different Krein signatures.