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Propagation of anisotropic Gabor singularities for Schrödinger type equations

Marco Cappiello, Luigi Rodino, Patrik Wahlberg

Abstract

We show results on propagation of anisotropic Gabor wave front sets for solutions to a class of evolution equations of Schrödinger type. The Hamiltonian is assumed to have a real-valued principal symbol with the anisotropic homogeneity $a(λx, λ^σξ) = λ^{1+σ} a(x,ξ)$ for $λ> 0$ where $σ> 0$ is a rational anisotropy parameter. We prove that the propagator is continuous on anisotropic Shubin--Sobolev spaces. The main result says that the propagation of the anisotropic Gabor wave front set follows the Hamilton flow of the principal symbol.

Propagation of anisotropic Gabor singularities for Schrödinger type equations

Abstract

We show results on propagation of anisotropic Gabor wave front sets for solutions to a class of evolution equations of Schrödinger type. The Hamiltonian is assumed to have a real-valued principal symbol with the anisotropic homogeneity for where is a rational anisotropy parameter. We prove that the propagator is continuous on anisotropic Shubin--Sobolev spaces. The main result says that the propagation of the anisotropic Gabor wave front set follows the Hamilton flow of the principal symbol.
Paper Structure (10 sections, 23 theorems, 260 equations)

This paper contains 10 sections, 23 theorems, 260 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Anisotropic Shubin pseudodifferential calculus
  4. Globally anisotropic Shubin--Sobolev spaces, localization operators and a sharp Gårding inequality
  5. Propagation of anisotropic Gabor wave front sets for evolution equations of Airy--Schrödinger type
  6. Fourier transformation of the evolution equation
  7. Anisotropically homogeneous Hamiltonians and their flows
  8. Solutions to a class of Schrödinger type equations with anisotropic Hamiltonians
  9. Propagation of anisotropic Gabor wave front sets for Schrödinger type equations
  10. Examples

Key Result

Lemma 3.2

If $m \in \mathbf R$, $\sigma,r > 0$ and $a \in C^\infty(\mathbf R^{2d})$ satisfies then $a \in G^{m,\sigma}$.

Theorems & Definitions (55)

  • Definition 3.1
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • Definition 3.4
  • Definition 3.5
  • Lemma 3.6
  • proof
  • Definition 4.1
  • Proposition 4.2
  • ...and 45 more