Exact Construction and Uniqueness of the Coupled-Channel Green's Function

TL;DR

This paper rigorously constructs and proves the uniqueness of the Green's function for coupled radial Schrödinger equations with symmetric potentials, applicable to nuclear, atomic, and molecular scattering.

Contribution

It provides a mathematically rigorous construction and proof of the unique Green's matrix for coupled Schrödinger equations with symmetric potentials, including applications in scattering theory.

Findings

The Wronskian matrix is diagonal with elements -k_n.

The construction is unique and satisfies boundary conditions and derivative discontinuity.

Application demonstrated in the continuum-discretized coupled-channels framework.

Abstract

We present a rigorous construction and uniqueness proof of the matrix Green's function for coupled radial Schr\"{o}dinger equations with symmetric coupling potentials. The Green's matrix $g_{\gamma\gamma'}(R,R')$ is built from two fundamental sets of $N$ linearly independent solutions, regular and outgoing, of the coupled radial equations. We prove that the associated Wronskian matrix is diagonal with elements $W_n = -k_n$ and independent of the radial coordinate, and demonstrate through the symplectic structure of the $2N$-dimensional phase space that the resulting construction is the unique Green's matrix satisfying the defining equation with correct boundary conditions, continuity at the source point, and the prescribed derivative discontinuity. The construction applies to any system of coupled radial Schr\"{o}dinger equations with symmetric coupling potentials and open channels, including coupled-channels problems arising in nuclear, atomic, and molecular scattering. As an illustrative application, we show how the Green's matrix enters the nonlocal dynamical polarization potential (DPP) within the continuum-discretized coupled-channels (CDCC) framework, where retaining the off-diagonal elements captures multistep excitation pathways beyond the weak-coupling approximation.