Cryptohermitian Picture of Scattering Using Quasilocal Metric Operators⋆

One-dimensional unitary scattering controlled by non-Hermitian (typically, PT -symmetric) quantum Hamiltonians H 6= 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 nonequivalent short-range metric operators Θ 6= I represented, in Runge–Kutta approximation, by (2R− 1)-diagonal matrices.