Raffaella Servadei, Enrico Valdinoci
On the spectrum of two different fractional operators
(539K, pdf)

ABSTRACT.  In this paper we deal with two nonlocal 
operators, 
that are both 
well known and widely 
studied in the literature 
in connection with elliptic problems of fractional type. 
Precisely, for a fixed $s\in (0,1)$ we consider the \emph{integral} definition 
of the fractional Laplacian given by 
$$(-\Delta)^s u(x):=rac{c(n,s)}{2}\int_{\RR^{n}}rac{2u(x)-u(x+y)-u(x-y)}{|y|^{n+2s}}\,dy\,,\,\,\,\, x\in \RR^n\,,$$ 
where $c(n,s)$ is a positive normalizing constant, and another fractional operator obtained via a 
\emph{spectral} definition, that is 
$$A_s u=\sum_{i\in \mathbb N}a_i\,\lambda_i^s\,e_i\,,$$ 
where 
$e_i\,, \lambda_i$ are the eigenfunctions and the eigenvalues of the Laplace operator $-\Delta$ in $\Omega$ with homogeneous Dirichlet boundary data, while $a_i$ represents the projection of $u$ on the direction $e_i$\,. 
Aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences.