  In this paper we discuss the topic of correct setting for the equation
$(-\Delta )^s u=f$, with $0<s <1$. The definition of the fractional Laplacian
on the whole space $\mathbb R^n$, $n=1,2,3$ is understood through the Fourier
transform, see, e.g., Lischke et.al. (J. Comp. Phys., 2020). The real challenge
however represents the case when this equation is posed in a bounded domain
$\Omega$ and proper boundary conditions are needed for the correctness of the
corresponding problem. Let us mention here that the case of inhomogeneous
boundary data has been neglected up to the last years. The reason is that
imposing nonzero boundary conditions in the nonlocal setting is highly
nontrivial. There exist at least two different definitions of fractional
Laplacian, and there is still ongoing research about the relations of them.
They are not equivalent. The focus of our study is a new characterization of
the spectral fractional Laplacian. One of the major contributions concerns the
case when the right hand side $f$ is a Dirac $\delta$ function. For comparing
the differences between the solutions in the spectral and Riesz formulations,
we consider an inhomogeneous fractional Dirichlet problem. The provided
theoretical analysis is supported by model numerical tests.
