  In this work we introduce volume constraint problems involving the nonlocal
operator $(-\Delta)_{\delta}^{s}$, closely related to the fractional Laplacian
$(-\Delta)^{s}$, and depending upon a parameter $\delta>0$ called horizon. We
study the associated linear and spectral problems and the behavior of these
volume constraint problems when $\delta\to0^+$ and $\delta\to+\infty$. Through
these limit processes on $(-\Delta)_{\delta}^{s}$ we derive spectral
convergence to the local Laplacian and to the fractional Laplacian as
$\delta\to 0^+$ and $\delta \to +\infty$ respectively, as well as we prove the
convergence of solutions of these problems to solutions of a local Dirichlet
problem involving $(-\Delta)$ as $\delta\to0^+$ or to solutions of a nonlocal
fractional Dirichlet problem involving $(-\Delta)^s$ as $\delta\to+\infty$.
