Discrete-coordinate crypto-Hermitian quantum system controlled by time-dependent Robin boundary conditions

Abstract

Non-stationary version of unitary quantum mechanics formulated in non-Hermitian (or, more precisely, in hiddenly Hermitian) interaction-picture representation is illustrated via an elementary $N$ by $N$ matrix Hamiltonian $H(t)$ mimicking a 1D-box system with physics controlled by time-dependent boundary conditions. The model is presented as analytically solvable at $N=2$. Expressis verbis, this means that for both of the underlying Heisenbergian and Schrödingerian evolution equations the generators (i.e., in our notation, the respective operators $\Sigma (t)$ and $G(t)$) become available in closed form. Our key message is that contrary to the conventional beliefs and in spite of the unitarity of the evolution of the system, neither its "Heisenbergian Hamiltonian" $\Sigma (t)$ nor its "Schrödingerian Hamiltonian" $G(t)$ possesses a real spectrum or the conjugate pairs of complex eigenvalues. This means that neither one of these "Hamiltonians" can be pseudo-Hermitian alias PT-symmetric.

arXiv comment19 pp., 1 figure

References in corpus (30)

• Making Sense of Non-Hermitian Hamiltonians
• Pseudo-Hermitian Representation of Quantum Mechanics
• Physical Aspects of Pseudo-Hermitian and $PT$-Symmetric Quantum Mechanics
• Time-dependent version of cryptohermitian quantum theory
• Unitary quantum evolution for time-dependent quasi-Hermitian systems with non-observable Hamiltonians
• Time Dependent $\mathcal{P}\mathcal{T}$-Symmetric Quantum Mechanics
• Three-Hilbert-Space Formulation of Quantum Mechanics
• Interface between Hermitian and non-Hermitian Hamiltonians in a model calculation
• On the invariant method for the time-dependent non-Hermitian Hamiltonians
• Quantum Mechanics of Klein-Gordon Fields I: Hilbert Space, Localized States, and Chiral Symmetry
• Exact analytical solutions for time-dependent Hermitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians
• Flattening the Curve with Einstein's Quantum Elevator: Hermitization of Non-Hermitian Hamiltonians via a Generalized Vielbein Formalism
• Non-Hermitian interaction representation and its use in relativistic quantum mechanics
• Perfect transmission scattering as a PT-symmetric spectral problem
• Calculation of the metric in the Hilbert space of a PT-symmetric model via the spectral theorem
• Quantum Mechanics of Proca Fields
• A Positive-Definite Scalar Product for Free Proca Particle
• Non-self-adjoint graphs
• Adiabatic time-dependent metrics in PT-symmetric quantum theories
• Crypto-unitary forms of quantum evolution operators
• An introduction to PT-symmetric quantum mechanics -- time-dependent systems
• Non-Hermitian coupled cluster method for non-stationary systems and its interaction-picture reinterpretation
• Spectrally equivalent time-dependent double wells and unstable anharmonic oscillators
• PT-symmetric quantum graphs
• Wheeler-DeWitt equation and the applicability of crypto-Hermitian interaction representation in quantum cosmology
• Solvable non-Hermitian discrete square well with closed-form physical inner product
• Quantum Graphs: $\mathcal{P}\mathcal{T}$-symmetry and reflection symmetry of the spectrum
• Composite quantum Coriolis forces
• Real energies and Berry phases in all PT-regimes in time-dependent non-Hermitian theories
• Non-Hermitian Quantum Fermi Accelerator

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• Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics
• Non-Hermitian Generalization of Rayleigh-Schrödinger Perturbation Theory
• Heisenberg and Heisenberg-Like Representations via Hilbert Space Bundle Geometry in the Non-Hermitian Regime
• Non-Hermitian-Hamiltonian-induced unitarity and optional physical inner products in Hilbert space
• Emergence and localization of exceptional points in an exactly solvable toy model