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Remark on Estimates in Modulation Spaces for Schrödinger Evolution Operators with Sub-quadratic Potentials

Kosuzu Hamaoka, Keiichi Kato, Shun Takizawa

Abstract

In this paper we give an estimate for the solution to the Schrödinger equation with sub-quadratic potentials in modulation spaces by the norm of the initial functions in Wiener-Amalgum spaces.

Remark on Estimates in Modulation Spaces for Schrödinger Evolution Operators with Sub-quadratic Potentials

Abstract

In this paper we give an estimate for the solution to the Schrödinger equation with sub-quadratic potentials in modulation spaces by the norm of the initial functions in Wiener-Amalgum spaces.
Paper Structure (7 sections, 3 theorems, 34 equations)

This paper contains 7 sections, 3 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Wave Packet Transform
  5. Modulation Spaces and Wiener-Amalgum spaces
  6. Key Lemmas
  7. Proof of Theorem \ref{['main-estimate']}

Key Result

Theorem 1.2

Suppose that $1\le p, q \le \infty$ satisfy Let $\varphi_0\in\mathcal{S}(\mathbb{R}^n)\backslash\{ 0\}$ and set $\varphi (t,x)=e^{\frac{1}{2}it\Delta} \varphi_0(x)$. If $V$ satisfies Assumption ass1-1, then there exist $T>0$ and $C>0$ such that for all $t\in [-T,T]$, where $u(t,x)$ denotes the solution of SE in $C(\mathbb{R} ;\mathcal{S}'(\mathbb{R}^n))$ with $u(0,x)=u_0(x)$.

Theorems & Definitions (5)

  • Theorem 1.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof : Proof of Theorem \ref{['main-estimate']}