$z$ is defined by $$\label{eq1,8}
P_z := \frac{1}{2i\pi} \oint_{\gamma} (T - \zeta)^{-1} d\zeta.$$ The algebraic multiplicity of $z$ is then defined by $$\label{eq1,9}
\textup{m}(z) := \text{rank} (P_z),$$ and when it is finite, the point $z$ is called a discrete eigenvalue of the operator $T$. Note that we have the inequality $\mathrm{m} \, (z) \geq \mathrm{dim} \, \big( \text{Ker} 
(T - z) \big)$, which is the geometric multiplicity of $z$. The equality holds if $T = T^\ast$. So, we define the discrete spectrum of $T$ as $$\label{eq1,10}
\sigma_{\text{\bf disc}}(T) := \big\lbrace z \in \sigma(T) : z \hspace*{0.1cm} 
\textup{is a discrete eigenvalue of $T$} \big\rbrace.$$ We recall that if a closed linear operator has a closed range and both its kernel and cokernel are finite-dimensional, then it is called a Fredholm operator. Hence, we define the essential spectrum of $T$ as $$\label{eq1,11}
\sigma_{\text{\bf ess}}(T) := \big\lbrace z \in {{\mathbb{C}}}: \textup{$T - z$ \textup{is not a 
Fredholm operator}} \big\rbrace.$$ Note that $\sigma_{\text{\bf ess}}(T)$ is a closed subset of $\sigma(T)$.

The paper is organized as follows. In Section \[secd\], we present our model. In section \[secp\], we state our main results Theorem \[t1\] and Corollaries \[t2\], \[t3\]. Section \[sec3\] is devoted to preliminary results 