Cryptohermitian Picture of Scattering Using Quasilocal Metric Operators

Details Cryptohermitian Picture of Scattering Using Quasilocal Metric Operators Cryptohermitian Picture of Scattering Using Quasilocal Metric Operators

2009-08-27

arxiv.org/abs/0908.4045v2

SIGMA 5 (2009), 085, 21 pages

Physics

Quantum Physics

Scientific paper

10.3842/SIGMA.2009.085

One-dimensional unitary scattering controlled by non-Hermitian (typically, ${\cal PT}$-symmetric) quantum Hamiltonians $H\neq H^\dagger$ 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 $\Theta\neq I$ represented, in Runge-Kutta approximation, by $(2R-1)$-diagonal matrices.

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