Hagedorn wavepackets in time-frequency and phase space

TL;DR

This paper explores Hagedorn wavepackets, a generalization of Hermite functions, analyzing their properties in phase space and their relation to other special functions, with explicit formulas for their transforms.

Contribution

It introduces explicit formulas and recurrence relations for Hagedorn wavepackets' transforms, connecting them to Laguerre polynomials and expanding their analytical framework.

Findings

Derived explicit formulas for Wigner and FBI transforms of wavepackets

Established relations between Hagedorn wavepackets and Laguerre polynomials

Enhanced understanding of phase space localization properties

Abstract

The Hermite functions are an orthonormalbasis of the space of square integrable functions with favourable approximation properties. Allowing for a flexible localization in position and momentum, the Hagedorn wavepackets generalize the Hermite functions also to several dimensions. Using Hagedorn's raising and lowering operators, we derive explicit formulas and recurrence relations for the Wigner and FBI transform of the wavepackets and show their relation to the Laguerre polyomials.