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.

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Cited by in corpus (2)

• Solvable non-Hermitian discrete square well with closed-form physical inner product
• PT-symmetric model with an interplay between kinematical and dynamical non-localities