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

TL;DR

This work extends the classical metaplectic calculus to the complex positive semigroup $Mp_+(d,\mathbb{C})$, providing a complete operator‑theoretic framework that parallels the real metaplectic theory but accommodates nonunitary evolutions. It develops generators and a polar decomposition, establishes a Schrödinger‑type intertwining for complex actions, and analyzes how complex conjugation interacts with metaplectic structure, all while tying these structural results to time‑frequency representations via metaplectic Wigner distributions. The paper then applies this framework to time‑frequency analysis—characterizing covariant representations and generalized spectrograms within the Cohen class and recovering Husimi/Gaussian instances—and to evolution equations with complex quadratic Hamiltonians, deriving modulation‑space bounds and microlocal propagation results for Wigner distributions. Overall, the theory provides a versatile, unified approach to phase‑space analysis and PDEs in the complex, positive‑definite setting, with potential impact on signal processing, quantum mechanics, and microlocal analysis of non‑selfadjoint operators, including explicit propagation of Wigner singularities. A key outcome is a robust link between complex metaplectic structure, time‑frequency representations, and evolution dynamics in a broad nonunitary regime.

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 Hörmander, 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.

Theorems & Definitions (105)

• Definition 1.1
• Definition 1.2
• Proposition 1.3
• Theorem 1.4
• Theorem 1.5
• Proposition 1.6
• Theorem 1.7
• Theorem 2.1
• Proposition 2.2
• Proposition 2.3