Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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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