Time delay and Calabi invariant in classical scattering theory

TL;DR

This paper develops a classical scattering theory framework, defining time delay and Calabi invariants, and provides explicit formulas and applications to various Hamiltonian systems, bridging classical and quantum scattering concepts.

Contribution

It introduces a classical version of the Eisenbud-Wigner formula and explicit expressions for the Calabi invariant derivative in Hamiltonian mechanics.

Findings

Explicit formulas for classical time delay in scattering

A classical Eisenbud-Wigner formula established

Application to dispersive Hamiltonians and Poincaré models

Abstract

We define, prove the existence and obtain explicit expressions for classical time delay defined in terms of sojourn times for abstract scattering pairs (H_0,H) on a symplectic manifold. As a by-product, we establish a classical version of the Eisenbud-Wigner formula of quantum mechanics. Using recent results of V. Buslaev and A. Pushnitski on the scattering matrix in Hamiltonian mechanics, we also obtain an explicit expression for the derivative of the Calabi invariant of the Poincar\'e scattering map. Our results are applied to dispersive Hamiltonians, to a classical particle in a tube and to Hamiltonians on the Poincar\'e ball.