An invariant class of wave packets for the Wigner transform

TL;DR

This paper demonstrates that the Wigner transform of generalized Hagedorn wave packets retains the same wave packet structure in phase space, revealing a tensor product structure and utilizing a novel Laguerre connection.

Contribution

It introduces a new analysis of the Wigner transform of Hagedorn wave packets, showing they form an invariant class in phase space with a novel Laguerre connection.

Findings

Wigner transform preserves the wave packet structure.

Identifies a tensor product structure in phase space.

Develops a new Laguerre connection for polynomial analysis.

Abstract

Generalised Hagedorn wave packets appear as exact solutions of Schr\"odinger equations with quadratic, possibly complex, potential, and are given by a polynomial times a Gaussian. We show that the Wigner transform of generalised Hagedorn wave packets is a wave packet of the same type in phase space. The proofs build on a parametrisation via Lagrangian frames and a detailed analysis of the polynomial prefactors, including a novel Laguerre connection. Our findings directly imply the recently found tensor product structure of the Wigner transform of Hagedorn wave packets.