Reciprocity Theorem and Fundamental Transfer Matrix

TL;DR

This paper introduces a new proof of the reciprocity theorem in multi-dimensional potential scattering using a fundamental transfer matrix, revealing operator identities and properties of the scattering operator applicable to complex potentials.

Contribution

It provides a novel proof of the reciprocity theorem in higher dimensions without relying on traditional Green's function methods, and generalizes the transfer matrix concept to multiple dimensions.

Findings

Identifies the fundamental transfer matrix as key to reciprocity.

Establishes an analog of $ ext{det} extbf{M} = 1$ in higher dimensions.

Reveals an anti-pseudo-Hermiticity property of the scattering operator.

Abstract

Stationary potential scattering admits a formulation in terms of the quantum dynamics generated by a non-Hermitian effective Hamiltonian. We use this formulation to give a proof of the reciprocity theorem in two and three dimensions that does not rely on the properties of the scattering operator, Green's functions, or Green's identities. In particular, we identify reciprocity with an operator identity satisfied by an integral operator $\widehat{\mathbf{M}}$, called the fundamental transfer matrix. This is a multi-dimensional generalization of the transfer matrix $\mathbf{M}$ of potential scattering in one dimension that stores the information about the scattering amplitude of the potential. We use the property of $\widehat{\mathbf{M}}$ that is responsible for reciprocity to identify the analog of the relation, $\det{\mathbf{M}}=1$, in two and three dimensions, and establish a generic anti-pseudo-Hermiticity of the scattering operator. Our results apply for both real and complex potentials.