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

  • * Corresponding author: Yong-Cheol Kim

    * Corresponding author: Yong-Cheol Kim
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  • 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 $\mathfrak{e}_V$.

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

    Citation:

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