Distributions of Off-Diagonal Scattering Matrix Elements: Exact Results

TL;DR

This paper derives exact distributions for the off-diagonal elements of the scattering matrix in chaotic systems, providing fundamental insights into universal scattering behaviors across various physical contexts.

Contribution

It presents the first exact analytical results for the distributions of off-diagonal scattering matrix elements for systems with orthogonal and unitary symmetry.

Findings

Exact distributions derived for off-diagonal scattering matrix elements

Applicable to systems modeled by the Heidelberg approach

Solves a longstanding problem in scattering theory

Abstract

Scattering is a ubiquitous phenomenon which is observed in a variety of physical systems which span a wide range of length scales. The scattering matrix is the key quantity which provides a complete description of the scattering process. The universal features of scattering in chaotic systems is most generally modeled by the Heidelberg approach which introduces stochasticity to the scattering matrix at the level of the Hamiltonian describing the scattering center. The statistics of the scattering matrix is obtained by averaging over the ensemble of random Hamiltonians of appropriate symmetry. We derive exact results for the distributions of the real and imaginary parts of the off-diagonal scattering matrix elements applicable to orthogonally-invariant and unitarily-invariant Hamiltonians, thereby solving a long standing problem.