Cryptohermitian Picture of Scattering Using Quasilocal Metric Operators

Abstract

One-dimensional unitary scattering controlled by non-Hermitian (typically, $\mathcal{P}\mathcal{T}$-symmetric) quantum Hamiltonians $H\ne H^{†}$ is considered. Treating these operators via Runge-Kutta approximation, our three-Hilbert-space formulation of quantum theory is reviewed as explaining the unitarity of scattering. Our recent paper on bound states [Znojil M., SIGMA 5 (2009), 001, 19 pages, arXiv:0901.0700] is complemented by the text on scattering. An elementary example illustrates the feasibility of the resulting innovative theoretical recipe. A new family of the so called quasilocal inner products in Hilbert space is found to exist. Constructively, these products are all described in terms of certain non-equivalent short-range metric operators $\Theta \ne I$ represented, in Runge-Kutta approximation, by $(2R-1)$-diagonal matrices.

References in corpus (26)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• The ODE/IM Correspondence
• On the reality of the eigenvalues for a class of PT-symmetric oscillators
• A family of complex potentials with real spectrum
• An Equivalent Hermitian Hamiltonian for the non-Hermitian -x^4 Potential
• Time-dependent version of cryptohermitian quantum theory
• Hilbert Space Structures on the Solution Space of Klein-Gordon Type Evolution Equations
• Metric Operator in Pseudo-Hermitian Quantum Mechanics and the Imaginary Cubic Potential
• Spectral Singularities, Biorthonormal Systems, and a Two-Parameter Family of Complex Point Interactions
• Ghost Busting: PT-Symmetric Interpretation of the Lee Model
• Three-Hilbert-Space Formulation of Quantum Mechanics
• Conceptual Problems in Scattering from Localized non-Hermitian Potentials
• Reflectionless Potentials and PT Symmetry
• Interface between Hermitian and non-Hermitian Hamiltonians in a model calculation
• Scattering theory with localized non-Hermiticities
• Handedness of complex PT-symmetric potential barriers
• The conditionally exactly solvable potentials: A misunderstanding (comment on "Conditionally exactly soluble class of quantum potentials")
• Discrete PT-symmetric models of scattering
• Cryptogauge symmetry and cryptoghosts for crypto-Hermitian Hamiltonians
• On the Path-Integral Derivation of the Anomaly for the Hermitian Equivalent of the Complex $PT$-Symmetric Quartic Hamiltonian
• Quantum Mechanics of Proca Fields
• A Positive-Definite Scalar Product for Free Proca Particle
• Adiabatic time-dependent metrics in PT-symmetric quantum theories
• Non-Hermitian Hamilton operator in open quantum systems
• Relativistic vector bosons and PT-symmetry

Cited by in corpus (3)

• A non self-adjoint model on a two dimensional noncommutative space with unbound metric
• Solvable non-Hermitian discrete square well with closed-form physical inner product
• PT-symmetric model with an interplay between kinematical and dynamical non-localities