Spectral properties of non- self-adjoint elliptic differential operators in the Hilbert space

Let $\Omega$ be a bounded domain in $R^{n}$ with smooth boundary $\partial\Omega$. In this article, we will investigate the spectral properties of a non-self adjoint elliptic differential operator\\ $(Au)(x)=-\sum^{n}_{i,j=1}\left(\omega^{2\alpha}(x)a_{ij}(x) \mu(x)u'_{x_{i}}(x)\right)'_{x_{j}}$, acting in the Hilbert space $H=L^{2}{(\Omega)}$. with Dirichlet-type boundary conditions. Here $a_{ij}(x)= \overline{a_{ji}(x)}\;\;\;(i,j=1,\ldots,n),\;\;\; a_{ij}(x)\in C^{2}(\overline{\Omega})$, and the functions $a_{ij}(x)$ satisfies the uniformly elliptic condition, and let $ 0 \leq \alpha < 1$. Furthermore, for $\forall x \in \overline{\Omega}$, the function $\mu(x)$ lie in the $\psi_{\theta_1\theta_2}$ , where ${\psi_{\theta_1\theta_2}}=\{z \in {\bf C}:\;\pi/2<\theta_1 \leq|arg\;z| \leq \theta_2<\pi\},$