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Wave front set of solutions to the fractional Schrödinger equation

Takumi Kanai, Ryo Muramatsu, Yuusuke Sugiyama

Abstract

In this paper, we characterize the wave front sets of solutions to fractional Schrödinger equations \(i\partial_{t}u =(-Δ)^{θ/2}u + V(x)u\) with $0<θ<2$ via the wave packet transform (short-time Fourier transform). We clarify the relationship between the order \(θ\) of the fractional Laplacian and the growth rate of the potential in the problem of propagation of singularities. In particular, we present a theorem that bridges the propagation mechanisms of singularities for the Schrödinger and wave equations.

Wave front set of solutions to the fractional Schrödinger equation

Abstract

In this paper, we characterize the wave front sets of solutions to fractional Schrödinger equations \(i\partial_{t}u =(-Δ)^{θ/2}u + V(x)u\) with via the wave packet transform (short-time Fourier transform). We clarify the relationship between the order of the fractional Laplacian and the growth rate of the potential in the problem of propagation of singularities. In particular, we present a theorem that bridges the propagation mechanisms of singularities for the Schrödinger and wave equations.
Paper Structure (8 sections, 5 theorems, 110 equations)

This paper contains 8 sections, 5 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Estimates for characteristic curves
  3. Representation of solutions via wave packet transform
  4. Proof of Main Theorem
  5. Proof of Corollaries
  6. Proof of Corollary \ref{['wfs:cor:main_no_potential']}
  7. Proof of Corollary \ref{['wfs:cor:main_time_propagation']} with $\theta=1$
  8. Proof of Corollary \ref{['wfs:cor:main_time_propagation']} with $0< \theta <1$

Key Result

Proposition 1.1

Let $(x_0, \xi_0) \in \mathbb{R}^n \times (\mathbb{R}^n \setminus \{0\})$ and $f \in \mathcal{S}'(\mathbb{R}^n)$. Fix $0 < b < 1$. Then, the following conditions are equivalent:

Theorems & Definitions (11)

  • Definition 1.1: Wave front set
  • Definition 1.2: Wave Packet Transform
  • Proposition 1.1: Kato, Kobayashi and Ito KatoKobayashiIto2017
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Corollary 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • ...and 1 more