The Malgrange-Ehrenpreis theorem for nonlocal Schr\"odinger operators with certain potentials

TL;DR

This paper proves the Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators with certain potentials, establishing the existence and decay properties of fundamental solutions in a generalized setting.

Contribution

It extends the Malgrange-Ehrenpreis theorem to nonlocal Schrödinger operators with nonnegative potentials in specific function spaces, providing fundamental solutions.

Findings

Existence of fundamental solutions for nonlocal Schrödinger operators.

Fundamental solutions satisfy the distributional equation with Dirac delta.

Decay properties of the fundamental solutions are established.

Abstract

In this paper, we prove the Malgrange-Ehrenpreis theorem for nonlocal Schr\"odinger operators $L_K+V$ with nonnegative potentials $V\in L^q_{\loc}(\BR^n)$ for $q>\f{n}{2s}$ with $0<s<1$ and $n\ge 2$; that is to say, we obtain the existence of a fundamental solution $\fe_V$ for $L_K+V$ satisfying \begin{equation*}\bigl(L_K+V\bigr)\fe_V=\dt_0\,\,\text{ in $\BR^n$ }\end{equation*} in the distribution sense, where $\dt_0$ denotes the Dirac delta mass at the origin. In addition, we obtain a decay of the fundamental solution $\fe_V$.