An invariant class of wave packets for the Wigner transform

Abstract

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.

The Hagedorn–Hermite Correspondence

“…In Section 5, we obtain the generating function for the Hagedorn wave packets and those polynomials appearing in them (called the Hagedorn polynomials in this paper) again exploiting the results from Section 3. Such a generating function is obtained by Dietert et al [5] and Hagedorn [14] using the recurrence relations and by direct calculations, respectively. Our approach is different from them in the sense that the generating function for the Hagedorn wave packets is obtained directly from that for the Hermite functions; particularly, this is done in a manner that exactly parallels the Hagedorn-Hermite correspondence obtained in Theorem 3.8.…”

“…As another example, this section takes the generating functions for the Hermite functions and polynomials and shows how they can be transformed into the generating functions for the Hagedorn wave packets and polynomials. Such generating functions are obtained in Dietert et al [5] and Hagedorn [14]. We present an alternative derivation of them based on Theorem 3.8 using the Heisenberg-Weyl and metaplectic operators.…”

Stable Interpolation with Isotropic and Anisotropic Gaussians Using Hermite Generating Function

“…Proof. We use the following anisotropic Hermite generating function (see [4,Lemma 5] or [14, Theorem 3.1] with A = Id d ):…”

“…where we used the Rodrigues formula for multivariate tensor-product Hermite polynomials (see [4,Expr. 11] with M = Id).…”

“…For certain classes of functions, anisotropic Gaussians yield improved accuracy as shown in [2]. To include these cases in our description, we use an anisotropic generating function recently obtained by Dietert, Keller, & Troppmann [4] as well as by Hagedorn [14] and generalize our HermiteGF expansion to the anisotropic case. We also propose and analyze a novel cut-off criterion, that contrary to the existing ones does not only account for the diagonal contributions of the stabilization but measures the impact of the full stabilization basis.…”