Spectrum for a weighted one-dimensional fractional Laplace operator

Abstract

In this paper, we study the spectrum of the one-dimensional fractional Laplace operator with a definite weight$\{\begin{array}{c}{(-\frac{d^2}{d{x}^2})}^{\alpha }u\left(x\right)=\lambda a\left(x\right)u\left(x\right),x\in (-1,1),\\ u\left(x\right)=0,x\text{in}R\setminus (-1,1),\end{array}$ where $\alpha \in (0,1),a\in C\left(\right[-1,1],[0,+\infty \left)\right)$ and ${(-\frac{d^2}{d{x}^2})}^{\alpha }$ is the one-dimensional fractional Laplace nonlocal operator. By virtue of Γ-convergence arguments, we investigate, in the singular limit, that the eigenvalue and eigenfunction of the nonlocal operator converge to those of the corresponding classical second-order two-point boundary value problem in the first place, and then, building upon the continuity of the eigenvalues and eigenfunctions as a function of fractional index, we derive the simplicity of the eigenvalues of the fractional Laplace nonlocal operator for all fractional index $\alpha \in (0,1)$ by adopting a stet-by-step iterative approach. Furthermore, using the α-harmonic extension, we receive that the corresponding eigenfunction ${\phi }_k^{\alpha }$ has at most $2k-2$ zeros in $(-1,1)$. At last, from an experiment point of view, we give the numerical eigenvalues and eigenfunctions of the weighted fractional Laplace problem by means of the finite element method in some special cases, which enriches the theoretical results. The spectral results presented here, especially the simplicity of eigenvalues ${\lambda }_k^{\alpha }$ resolve a conjecture of Bañuelos and Kulczycki.