Characterization of modulation spaces by symplectic representations and applications to Schr\"{o}dinger equations

TL;DR

This paper explores how symplectic representations and metaplectic operators can characterize modulation spaces and be applied to Schrödinger equations, offering a new perspective in time-frequency analysis and quantum mechanics.

Contribution

It introduces a symplectic perspective on time-frequency representations, establishing conditions for their equivalence to the STFT in defining modulation spaces and applying them to Schrödinger equations.

Findings

Symplectic time-frequency representations can replace the STFT for modulation spaces.

Conditions are identified under which these representations are equivalent to the STFT.

Application of symplectic representations to Schrödinger equations demonstrates their practical utility.

Abstract

In the last twenty years modulation spaces, introduced by H. G. Feichtinger in 1983, have been successfully addressed to the study of signal analysis, PDE's, pseudodifferential operators, quantum mechanics, by hundreds of contributions. In 2011 M. de Gosson showed that the time-frequency representation Short-time Fourier Transform (STFT), which is the tool to define modulation spaces, can be replaced by the Wigner distribution. This idea was further generalized to $\tau$-Wigner representations in [9]. In this paper time-frequency representations are viewed as images of symplectic matrices via metaplectic operators. This new perspective highlights that the protagonists of time-frequency analysis are metaplectic operators and symplectic matrices $\mathcal{A} \in Sp(2d,\mathbb{R})$. We find conditions on $\mathcal{A}$ for which the related symplectic time-frequency representation $W_\mathcal{A}$ can replace the STFT and give equivalent norms for weighted modulation spaces. In particular, we study the case of covariant matrices $\mathcal{A}$, i.e., their corresponding $W_\mathcal{A}$ are members of the Cohen class. Finally, we show that symplectic time-frequency representations $W_\mathcal{A}$ can be efficiently employed in the study of Schr\"{o}dinger equations. This new approach may have further applications in quantum mechanics and PDE's.