Semi-classical Time-frequency Analysis and Applications

TL;DR

This paper develops a unified framework combining semi-classical and time-frequency analysis to study Schrödinger equations, establishing continuity and asymptotic results on $ ext{-dependent Banach spaces with operator norms independent of .

Contribution

It introduces a novel approach linking semi-classical analysis with time-frequency methods, including a generalized Gabor frame, for Schrödinger operators.

Findings

Continuity of Schrödinger propagators on -dependent Banach spaces.

Operator norms are independent of Planck's constant .

Extension of analysis to Fourier integral operators and their sparsity.

Abstract

This work represents a first systematic attempt to create a common ground for semi-classical and time-frequency analysis. These two different areas combined together provide interesting outcomes in terms of Schr\"odinger type equations. Indeed, continuity results of both Schr\"odinger propagators and their asymptotic solutions are obtained on $\hbar$-dependent Banach spaces, the semi-classical version of the well-known modulation spaces. Moreover, their operator norm is controlled by a constant independent of the Planck's constant $\hbar$. The main tool in our investigation is the joint application of standard approximation techniques from semi-classical analysis and a generalized version of Gabor frames, dependent of the parameter $\hbar$. Continuity properties of more general Fourier integral operators (FIOs) and their sparsity are also investigated.