The Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators with certain potentials

Woocheol Choi; Yong-Cheol Kim

doi:10.3934/cpaa.2018095

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2018, Volume 17, Issue 5: 1993-2010. Doi: 10.3934/cpaa.2018095

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The Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators with certain potentials

Woocheol Choi^1, and
Yong-Cheol Kim^2, ,

1.
Department of Mathematics Education, Incheon National University, Incheon 22012, Republic of Korea
2.
Department of Mathematics Education, Korea University, Seoul 02841, Republic of Korea

* Corresponding author: Yong-Cheol Kim
* Corresponding author: Yong-Cheol Kim

Received: July 31, 2017

Revised: December 31, 2017

Published: April 2018

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Abstract

In this paper, we prove the Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators $L_K+V$ with nonnegative potentials $V∈ L^q_{\rm{loc}}(\mathbb{R}^n)$ for $q>\frac{n}{2s}$ with $0 < s < 1$ and $n>2s$; that is to say, we obtain the existence of a fundamental solution $\mathfrak{e}_V$ for $L_K+V$ satisfying

$\begin{equation*}\bigl(L_K+V\bigr)\mathfrak{e}_V = \delta _0\,\,\text{ in $\mathbb{R}^n$ }\end{equation*}$

in the distribution sense, where $\delta _0$ denotes the Dirac delta mass at the origin. In addition, we obtain a decay of the fundamental solution $\mathfrak{e}_V$.

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Mathematics Subject Classification: Primary: 47G20, 45K05, 35J60, 35B65, 35D30; Secondary: 60J75.

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