I guess this topic may have been addressed somewhere but I cannot really find a reference myself, so I ask here.

For each $N \in \mathbb{N}$, consider the Laplacian $\Delta$ on the interval $[-N,N]$ subject to the periodic boundary conditions. Then, it is well-known that it is a self-adjoint operator on $L^2[-N,N]$ with discrete spectrum. Yet, the "gaps" between neighboring eigenvalues become "smaller" with larger $N$.

Now, the Laplacian $\Delta$ on whole $\mathbb{R}$ has the continuous spectrum $(-\infty,0]$. 

My question is that:
\begin{equation}
\text{ In what sense does the Laplacian } \Delta \text{ on } [-N,N] \text{ converges to one on the whole real line as } N \to \infty? 
\end{equation}
\begin{equation}
\text{ Moreover, does the boundary conditions on } [-N,N] \text{ matter as long as } \Delta \text{ remains self-adjoint?}
\end{equation}

Could anyone please clarify for me?