Hagedorn Wavepackets in Time-Frequency and Phase Space

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.