The fractional Schr\"odinger equation with general nonnegative potentials. The weighted space approach

TL;DR

This paper investigates the well-posedness and boundary behavior of solutions to the fractional Schrödinger equation with general nonnegative potentials, including super singular cases, using a weighted space framework.

Contribution

It introduces a novel weighted space approach to establish well-posedness for very weak solutions with broad classes of data and potentials.

Findings

Established well-posedness for the fractional Schrödinger equation with general nonnegative potentials.

Analyzed boundary behavior of solutions, including super singular potentials near the boundary.

Provided insights into the properties of solutions under various potential singularities.

Abstract

We study the Dirichlet problem for the stationary Schr\"odinger fractional Laplacian equation $(-\Delta)^s u + V u = f$ posed in bounded domain $ \Omega \subset \mathbb R^n$ with zero outside conditions. We consider general nonnegative potentials $V\in L^1_{loc}(\Omega)$ and prove well-posedness of very weak solutions when the data are chosen in an optimal class of weighted integrable functions $f$. Important properties of the solutions, such as its boundary behaviour, are derived. The case of super singular potentials that blow up near the boundary is given special consideration. Related literature is commented.