## Differential and Integral Equations

- Differential Integral Equations
- Volume 28, Number 11/12 (2015), 1085-1104.

### Scattering operator for semirelativistic Hartree type equation with a short range potential

Nakao Hayashi, Pavel I. Naumkin, and Takayoshi Ogawa

#### Abstract

We prove the existence of the scattering operator in the neighborhood of the origin in $\mathbf{H}^{\omega ,1}\cap \mathbf{H}^{\mu },$ where $\mu > \omega =1+\frac{2}{n}$ for the semirelativistic Hartree type equation \begin{equation*} i\partial _{t}u=\sqrt{M^{2}-\Delta }u+F \big ( u \big ) ,\text{ } \big ( t,x \big ) \in {\mathbf{R}}\times {\mathbf{R}}^{n}, \end{equation*} where $F ( u ) = ( | x | ^{-\gamma }\ast | u | ^{2} ) u,1 < \gamma < 2,n\geq 3$ and $\ast $ denotes the convolution.

#### Article information

**Source**

Differential Integral Equations, Volume 28, Number 11/12 (2015), 1085-1104.

**Dates**

First available in Project Euclid: 18 August 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1439901043

**Mathematical Reviews number (MathSciNet)**

MR3385136

**Zentralblatt MATH identifier**

1374.35330

**Subjects**

Primary: 35Q40: PDEs in connection with quantum mechanics 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 47J35: Nonlinear evolution equations [See also 34G20, 35K90, 35L90, 35Qxx, 35R20, 37Kxx, 37Lxx, 47H20, 58D25]

#### Citation

Hayashi, Nakao; Naumkin, Pavel I.; Ogawa, Takayoshi. Scattering operator for semirelativistic Hartree type equation with a short range potential. Differential Integral Equations 28 (2015), no. 11/12, 1085--1104. https://projecteuclid.org/euclid.die/1439901043