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

Abstract

‎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 ${\psi_{\theta_1\theta_2}}=\{z \in‎ ‎{\bf C}:\;\pi/2<\theta_1 \leq|arg\;z| \leq \theta_2<\pi\},$‎