  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.
