 12123 Raffaella Servadei, Enrico Valdinoci
 On the spectrum of two different fractional operators
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Oct 19, 12

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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
$$(\Delta)^s u(x):=rac{c(n,s)}{2}\int_{\RR^{n}}rac{2u(x)u(x+y)u(xy)}{y^{n+2s}}\,dy\,,\,\,\,\, x\in \RR^n\,,$$
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.
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