Effective Schroedinger equations for nonlocal and/or dissipative systems

TL;DR

This paper generalizes the projection formalism to derive effective Schrödinger equations applicable to nonlocal, nonhermitian, and dissipative systems, enabling practical calculations without explicit projection operators.

Contribution

It introduces a generalized formalism for effective Schrödinger equations that handles nonlocal, nonhermitian, and dissipative systems, simplifying practical computations.

Findings

Unified framework for nonlocal and dissipative systems

Formulas applicable without explicit projection operators

Enhanced methods for multichannel scattering calculations

Abstract

The projection formalism for calculating effective Hamiltonians and resonances is generalized to the nonlocal and/or nonhermitian case, so that it is applicable to the reduction of relativistic systems (Bethe-Salpeter equations), and to dissipative systems modeled by an optical potential. It is also shown how to recover all solutions of the time-independent Schroedinger equation in terms of solutions of the effective Schroedinger equation in the reduced state space and a Schroedinger equation in a reference state space. For practical calculations, it is important that the resulting formulas can be used without computing any projection operators. This leads to a modified coupled reaction channel/resonating group method framework for the calculation of multichannel scattering information.