Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the microlocal phase space concentration of Schrödinger evolutions

Gianluca Giacchi, Davide Tramontana

TL;DR

This work develops a comprehensive microlocal framework for Schrödinger evolutions using metaplectic Wigner distributions. It introduces $ ext{A}$-Wigner kernels and the $ ext{A}$-Wigner wave front set, establishing microlocality results for Weyl operators and FIOs with quadratic phase, and analyzes interactions between two evolving states through cross $ ext{A}$-Wigner front sets, highlighting ghost frequencies. By extending microlocal propagation from Weyl symbols to general metaplectic representations, the authors unify time-frequency and microlocal perspectives and provide precise phase-space inclusion results for both single-state dynamics and two-state interactions. The findings offer refined tools for tracking phase-space concentration and interference in quantum evolutions, with potential implications for quantum mechanics and signal analysis. The work also clarifies how Gaussian smoothing connects Wigner-based methods to the classical Gabor/WF framework, effectively bridging two complementary microlocal paradigms.

Abstract

In this work, we investigate the microlocal properties of the evolutions of Schrödinger equations using metaplectic Wigner distributions. So far, only restricted classes of metaplectic Wigner distributions, satisfying particular structural properties, have allowed the analysis of microlocal properties. We first extend the microlocal results to all metaplectic Wigner distributions, including the well-known Kohn-Nirenberg quantization, and examine these findings in the framework of Fourier integral operators with quadratic phase. Finally, we analyze the phase space concentration of the (cross) Wigner distribution arising from the interaction of two states, with particular attention to interactions generated by certain Schrödinger evolutions. These contributions enable a more refined study of the so-called ghost frequencies.

On the microlocal phase space concentration of Schrödinger evolutions

TL;DR

This work develops a comprehensive microlocal framework for Schrödinger evolutions using metaplectic Wigner distributions. It introduces -Wigner kernels and the -Wigner wave front set, establishing microlocality results for Weyl operators and FIOs with quadratic phase, and analyzes interactions between two evolving states through cross -Wigner front sets, highlighting ghost frequencies. By extending microlocal propagation from Weyl symbols to general metaplectic representations, the authors unify time-frequency and microlocal perspectives and provide precise phase-space inclusion results for both single-state dynamics and two-state interactions. The findings offer refined tools for tracking phase-space concentration and interference in quantum evolutions, with potential implications for quantum mechanics and signal analysis. The work also clarifies how Gaussian smoothing connects Wigner-based methods to the classical Gabor/WF framework, effectively bridging two complementary microlocal paradigms.

Abstract

In this work, we investigate the microlocal properties of the evolutions of Schrödinger equations using metaplectic Wigner distributions. So far, only restricted classes of metaplectic Wigner distributions, satisfying particular structural properties, have allowed the analysis of microlocal properties. We first extend the microlocal results to all metaplectic Wigner distributions, including the well-known Kohn-Nirenberg quantization, and examine these findings in the framework of Fourier integral operators with quadratic phase. Finally, we analyze the phase space concentration of the (cross) Wigner distribution arising from the interaction of two states, with particular attention to interactions generated by certain Schrödinger evolutions. These contributions enable a more refined study of the so-called ghost frequencies.

Paper Structure

This paper contains 20 sections, 35 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Wigner distribution
  4. Symplectic matrices
  5. Metaplectic operators
  6. Metaplectic Wigner distributions
  7. Pseudodifferential operators
  8. Hamilton map
  9. $\mathcal{A}$-Wigner kernels
  10. Definition of $\mathcal{A}$-Wigner kernels and main properties
  11. Doubling the variables
  12. $\mathcal{A}$-Wigner microlocal analysis
  13. The $\mathcal{A}$-Wigner operator $K_\mathcal{A}$ and the Weyl quantization
  14. The $\mathcal{A}$-Wigner wave front set
  15. $\mathcal{A}$-Wigner microlocality for pseudodifferential operators
  16. ...and 5 more sections

Key Result

Theorem 1.3

cordero2024wigner Let $W_\mathcal{A}$ be a shift-invertible and covariant metaplectic Wigner distribution. Then, for every $a\in S^0_{0,0}(\mathbb{R}^{2n})$ and every $f\in L^2(\mathbb{R}^n)$,

Theorems & Definitions (79)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 69 more