Summary: Let \(\Omega\) be a bounded domain in \(\mathbb R^n\) with smooth boundary \(\partial\Omega\). In this paper, we investigate the spectral properties of a non-selfadjoint elliptic differential operator \((Au)(x)= -\sum_{i,j=1}^n (\rho^{2\alpha}(x) a_{ij}(x)q(x) u_{x_i}'(x))_{x_j}'\) acting on Hilbert space \(H=L^2(\Omega)\) with Dirichlet-type boundary conditions. Here, \(\rho(x)= \text{dist}\{x,\partial\Omega\}\), \(0\leq\alpha<1\), \(q(x)\in C^2(\overline{\Omega})\), \(a_{ij}(x)\in C^2(\overline{\Omega})\), \(a_{ij}(x)= a_{ji}(x)\), and there exists \(c>0\) such that \(c|s|^2\leq \sum_{i,j=1}^n a_{ij}(x) s_i\overline{s_j}\) \((s=(s_1,\dots,s_n)\in\mathbb C^n\), \(x\in\Omega)\). We assume that for all \(x\in\overline{\Omega}\), \(q(x)\in\mathbb C\setminus\Phi\), where \(\Phi= \{z\in\mathbb C:|\arg z|\leq\varphi\}\), \(\varphi\in (0,\pi)\).