The metaplectic semigroup and its applications to time-frequency analysis and evolution operators

TL;DR

This paper extends the classical metaplectic theory to a broader complex symplectic setting, providing new operator-theoretic insights into the structure and applications of the metaplectic semigroup in time-frequency analysis.

Contribution

It introduces a systematic analysis of the metaplectic semigroup for complex symplectic matrices, expanding beyond quadratic evolution equations and classical unitary cases.

Findings

Characterizes generators, polar decomposition, and intertwining relations of the metaplectic semigroup.

Provides structural insights into classes of time-frequency representations with specific properties.

Studies boundedness and estimates of propagators for complex quadratic Hamiltonians on modulation spaces.

Abstract

We develop a systematic analysis of the metaplectic semigroup $\mathrm{Mp}_+(d,\mathbb{C})$ associated with positive complex symplectic matrices, a notion introduced almost simultaneously and independently by H\"ormander, Brunet, Kramer, and Howe, thereby extending the classical metaplectic theory beyond the unitary setting. While the existing literature has largely focused on propagators of quadratic evolution equations, for which results are typically obtained via Mehler formulas, our approach is operator-theoretic and symplectic in spirit and adapts techniques from the standard metaplectic group $\mathrm{Mp}(d,\mathbb{R})$ to a substantially broader framework that is not driven by differential problems or particular propagators. This point of view provides deeper insight into the structure of the metaplectic semigroup, and allows us to investigate its generators, polar decomposition, and intertwining relations with complex conjugation and with the Wigner distribution. We then exploit these structural results to characterize, from a metaplectic perspective, classes of time-frequency representations satisfying prescribed structural properties. Finally, we discuss further implications for parabolic equations with complex quadratic Hamiltonians, we study the boundedness of their propagators on modulation spaces, we obtain estimates in time of their operator norms. Finally, we apply our theory to the study of propagation of Wigner singularities.