  In this article we study some spectral properties of the linear operator
$\mathcal{L}\_{\Omega}+a$ defined on the space $C(\bar\Omega)$ by :$$
\mathcal{L}\_{\Omega}[\varphi]
+a\varphi:=\int\_{\Omega}K(x,y)\varphi(y)\,dy+a(x)\varphi(x)$$ where
$\Omega\subset \mathbb{R}^N$ is a domain, possibly unbounded, $a$ is a
continuous bounded function and $K$ is a continuous, non negative kernel
satisfying an integrability condition. We focus our analysis on the properties
of the generalised principal eigenvalue $\lambda\_p(\mathcal{L}\_{\Omega}+a)$
defined by $$\lambda\_p(\mathcal{L}\_{\Omega}+a):= \sup\{\lambda \in \mathbb{R}
\,|\, \exists \varphi \in C(\bar \Omega), \varphi\textgreater{}0, \textit{such
that}\, \mathcal{L}\_{\Omega}[\varphi] +a\varphi +\lambda\varphi \le 0 \,
\text{in}\;\Omega\}. $$ We establish some new properties of this generalised
principal eigenvalue $\lambda\_p$. Namely, we prove the equivalence of
different definitions of the principal eigenvalue. We also study the behaviour
of $\lambda\_p(\mathcal{L}\_{\Omega}+a)$ with respect to some scaling of $K$.
For kernels $K$ of the type, $K(x,y)=J(x-y)$ with $J$ a compactly supported
probability density, we also establish some asymptotic properties of
$\lambda\_{p} \left(\mathcal{L}\_{\sigma,m,\Omega}
-\frac{1}{\sigma^m}+a\right)$ where $\mathcal{L}\_{\sigma,m,\Omega}$ is defined
by
$\displaystyle{\mathcal{L}\_{\sigma,m,\Omega}[\varphi]:=\frac{1}{\sigma^{2+N}}\int\_{\Omega}J\left(\frac{x-y}{\sigma}\right)\varphi(y)\,
dy}$. In particular, we prove that $$\lim\_{\sigma\to
0}\lambda\_p\left(\mathcal{L}\_{\sigma,2,\Omega}-\frac{1}{\sigma^{2}}+a\right)=\lambda\_1\left(\frac{D\_2(J)}{2N}\Delta
+a\right),$$where $D\_2(J):=\int\_{\mathbb{R}^N}J(z)|z|^2\,dz$ and $\lambda\_1$
denotes the Dirichlet principal eigenvalue of the elliptic operator. In
addition, we obtain some convergence results for the corresponding
eigenfunction $\varphi\_{p,\sigma}$.
