## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Asymptotic Analysis for Nonlinear Dispersive and Wave Equations, K. Kato, T. Ogawa and T. Ozawa, eds. (Tokyo: Mathematical Society of Japan, 2019), 389 - 400

### Kato type smoothing estimates for magnetic Schrödinger equations with rough potentials

#### Abstract

We consider the magnetic Schrödinger equation $iD_0 u = D_k D^k u$ with electro-magnetic potential $A = (A_\mu)$ in $\boldsymbol{R}^{1+n}$, where $n \ge 3$ and $D_\mu = \partial_\mu + iA_\mu$. Under the assumption $A \in \cap_{j=0}^1 C^j(0,T;H^{1-j}_n)$ with $\sum_\alpha \| F_{\mu\nu};L^\infty (0,T; L^n(Q_\alpha))\|^2 \lt \infty$, we prove the Kato type smoothing estimate $$\sup_\alpha \| u; L^2(0,T; H^{1/2} (Q_\alpha)\| \lesssim \| u(0,\cdot)\|_2,$$ where $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ and $\{Q_\alpha\}_{\alpha \in \boldsymbol{Z}^n}$ is the family of unit cubes.

#### Article information

**Dates**

Received: 20 December 2015

Revised: 15 July 2016

First available in Project Euclid:
31 October 2019

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1572545253

**Digital Object Identifier**

doi:10.2969/aspm/08110389

**Subjects**

Primary: 35Q40: PDEs in connection with quantum mechanics 35Q41: Time-dependent Schrödinger equations, Dirac equations

**Keywords**

Smoothing effect Schrödinger equations Electro-magnetic potentials

#### Citation

Wada, Takeshi. Kato type smoothing estimates for magnetic Schrödinger equations with rough potentials. Asymptotic Analysis for Nonlinear Dispersive and Wave Equations, 389--400, Mathematical Society of Japan, Tokyo, Japan, 2019. doi:10.2969/aspm/08110389. https://projecteuclid.org/euclid.aspm/1572545253