We investigate the time evolution of Hagedorn wavepackets by non-Hermitian quadratic Hamiltonians. We state a direct connection between coherent states and Lagrangian frames. For the time evolution a multivariate polynomial recursion is derived that describes the activation of lower lying excited states, a phenomenon unprecedented for Hermitian propagation. Finally we apply the propagation of excited states to the Davies-Swanson oscillator.Re(ZZ * )ΩZ = iZ , Re(ZZ * )ΩZ = −iZ , so that (Re(ZZ * )Ω) 2 = −Id 2n .Proof. Writing π L + πL = Id 2n in terms of Z, we obtain − Im(ZZ * )Ω T = Id 2n . Hence, Im(ZZ * ) = Ω T . This implies symplecticity of the real part, sinceChecking positive definiteness, we see z · Re(ZZ * )z = 1 2 z · (ZZ * z +ZZ T z) = |Z * z| ≥ 0

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.

Generalised Hagedorn wave packets appear as exact solutions of Schrödinger 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.

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