PT-symmetry entails pseudo-Hermiticity regardless of diagonalizability

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.

arXiv comment11pages, 1 figure. arXiv admin note: text overlap with arXiv:1801.01676

References in corpus (16)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities
• Complex Extension of Quantum Mechanics
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• PT-symmetry breaking and laser-absorber modes in optical scattering systems
• Visualization of Branch Points in PT-Symmetric Waveguides
• Pseudo-Hermitian Representation of Quantum Mechanics
• Experimental Study of Active LRC Circuits with PT-Symmetries
• Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries
• PT-symmetry in honeycomb photonic lattices
• The ODE/IM Correspondence
• Quantum noise and self-sustained radiation of PT-symmetric systems
• The energy spectrum of complex periodic potentials of the Kronig-Penney type
• Statistical PT-symmetric lasing in an optical fiber network
• Kelvin-Helmholtz instability is the result of parity-time symmetry breaking
• On the structure of the two-stream instability -- complex G-Hamiltonian structure and Krein collisions between positive- and negative-action modes

Cited by in corpus (21)

• Non-Hermitian Photonic Lattices: tutorial
• $\mathcal{P}\mathcal{T}$-Symmetric Topological Edge-Gain Effect
• Always-Real-Eigenvalued Non-Hermitian Topological Systems
• Degeneracies and symmetry breaking in pseudo-Hermitian matrices
• Spontaneous and explicit parity-time-symmetry breaking in drift wave instabilities
• Fate of Topological Edge States in Disordered Periodically-driven Nonlinear Systems
• Formation of Exceptional Points in pseudo-Hermitian Systems
• Braiding topology of symmetry-protected degeneracy points in non-Hermitian systems
• Operator size growth in Lindbladian SYK
• Topological modes and spectral flows in inhomogeneous PT-symmetric continuous media
• Electric conductivity in non-Hermitian holography
• Entropy and entanglement in a bipartite quasi-Hermitian system and its Hermitian counterparts
• Retardaion-induced exceptional point
• Essential implications of similarities in non-Hermitian systems
• Natural polynomials for Kerr quasi-normal modes
• Quantum Instability
• Quantum implementation of non-unitary operations with biorthogonal representations
• Wave Topology in Hall MHD
• The analytically tractable zoo of similarity-induced exceptional structures
• Interferometric Geometric Phases of $\mathcal{P}\mathcal{T}$-symmetric Quantum Mechanics
• Edge-wave phase-shifts versus normal-mode phase-tilts in an Eady problem with a sloping boundary