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SIGMA 5 (2009), 085, 21 pages arXiv:0908.4045
https://doi.org/10.3842/SIGMA.2009.085
Contribution to the Proceedings of the 5-th Microconference Analytic and Algebraic Methods V
Cryptohermitian Picture of Scattering Using Quasilocal Metric Operators
Miloslav Znojil
Nuclear Physics Institute ASCR, 250 68 Rez, Czech Republic
Received July 05, 2009, in final form August 23, 2009; Published online August 27, 2009
Abstract
One-dimensional unitary scattering controlled by
non-Hermitian (typically, PT-symmetric) quantum
Hamiltonians H ≠ 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
Θ ≠ I represented, in Runge-Kutta approximation, by
(2R–1)-diagonal matrices.
Key words:
cryptohermitian observables; unitary scattering; Runge-Kutta discretization; quasilocal metric operators.
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