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

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.

arXiv comment28 pp., 4 tables, 1 figure

References in corpus (24)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Making Sense of Non-Hermitian Hamiltonians
• 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-Symmetric Quantum Mechanics
• Pseudo-Hermiticity versus PT-Symmetry II: A complete characterizatio n of non-Hermitian Hamiltonians with a real spectrum
• Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries
• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• Physical Aspects of Pseudo-Hermitian and $PT$-Symmetric Quantum Mechanics
• Supersymmetry and the spontaneous breakdown of PT symmetry
• On the reality of the eigenvalues for a class of PT-symmetric oscillators
• Weak pseudo-Hermiticity and antilinear commutant
• Delta-Function Potential with a Complex Coupling
• Conditions for complex spectra in a class of PT symmetric potentials
• Spontaneous breakdown of PT symmetry in the solvable square-well model
• Maximal couplings in PT-symmetric chain-models with the real spectrum of energies
• Construction of a unique metric in quasi-Hermitian quantum mechanics: non-existence of the charge operator in a 2 x 2 matrix model
• Krein space related perturbation theory for MHD alpha-2-dynamos and resonant unfolding of diabolical points
• A return to observability near exceptional points in a schematic PT-symmetric model
• Beyond the WKB approximation in PT-symmetric quantum mechanics
• Conditional observability
• A note on the integrability of non-Hermitian extensions of Calogero-Moser-Sutherland models
• PT-Invariant Periodic Potentials with a Finite Number of Band Gaps
• Determination of the domain of the admissible matrix elements in the four-dimensional PT-symmetric anharmonic model

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