Cryptohermitian Picture of Scattering Using Quasilocal Metric Operators

TL;DR

This paper develops a novel theoretical framework for one-dimensional scattering with non-Hermitian Hamiltonians using quasilocal metric operators, ensuring unitarity through a three-Hilbert-space formulation and approximations.

Contribution

It introduces a new family of quasilocal inner products in Hilbert space described by short-range metric operators in the context of non-Hermitian scattering theory.

Findings

Existence of a new family of quasilocal inner products.

Representation of metric operators as (2R-1)-diagonal matrices.

Feasibility demonstrated through an elementary example.

Abstract

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.