multiplication operator $B_i$, and the domain $D_i$ $(i=1,2,3)$ as follows, $$\label{Ki}
K_i: \mathcal X_i\to\mathcal X_i,\,\, K_iu(t, x)=
\begin{cases}
\int_D \kappa(y-x)u(t, y)dy\quad &\forall u\in X_i,\quad i=1,2,\\
\int_{{\ensuremath{\mathbb{R}}}^N}\kappa(y-x)u(t, y)dy\quad&\forall u\in X_3,
\end{cases}$$ \[bi\] B\_i: X\_iX\_i, B\_iu=b\_i u   b\_i(x)=

1 & i=1, 3,\
\_D(y-x)dy & i=2,

and $$\label{D-i}
D_i=\begin{cases}  D\quad &{\rm for}\,\, i=1,2,\cr
[0,p_1]\times[0,p_2]\times [0,p_N] &{\rm for}\,\, i=3.
\end{cases}$$ For $1\leq i\leq 3$, set \[Li\] (L\_iu)(t, x)=-\_tu(t, x)+(K\_iu)(t, x)-(B\_iu)(t, x) with domain $$\mathcal D (\mathcal L_i)=\{u\in \mathcal X_i| u \text{ is } C^1 \text{ in } t \text{ and } u_t\in \mathcal X_i\}.$$ Then for $i=1,2,3$, the eigenvalue problems , , and in the space $\mathcal X_i$ can be written uniformly as $$\label{EP-lambdai}
\mathcal L_iu+\lambda_i m_i u=0,$$ Observe that $K_2=K_1$, $B_3=B_1$ and $D_2=D_1$. The introduction of $K_2$, $B_3$, and $D_2$ is for convenience.

For any given real number $\lambda_i$, the weighted eigenvalue problem is closely related to the following regular eigenvalue problem on $\mathcal X_i$, \[EP-mui\] L\_i u+\_i m\_iu=\_i u. Let $\sigma_i(\mathcal{L}_i+\lambda_i m_i)$ be the spectrum set of (i.e. the spectrum set of the operator $\mathcal{L}_i+\lambda_im_i$ in $\mathcal{X}_i$) and $$\ds\mu_i^n(\mathcal{L}_i+\lambda_i m_i)=\sup\{{\rm Re}\mu|\mu\in\sigma_i(\mathcal{L}_i+\lambda_i m_i)\}.$$

\[PSP-def\]

-   $\mu_i^n(\mathcal{L}_i+\lambda_i m_i)$ is called [*the principal spectrum point*]{} of the regular eigenvalue problem . The principal spectrum point 