Spectral theory of elliptic differential operators with indefinite   weights
  The spectral properties of a class of non-selfadjoint second order elliptic operators with indefinite weight functions on unbounded domains $\Omega$ are investigated. It is shown that under an abstract regularity assumption the nonreal spectrum of the associated elliptic operator in $L^2(\Omega)$ is bounded. In the special case that $\Omega=R^n $decomposes into subdomains $\Omega_+$ and $\Omega_-$ with smooth compact boundaries and the weight function is positive on $\Omega_+$ and negative on $\Omega_-$, it turns out that the nonreal spectrum consists only of normal eigenvalues which can be characterized with a Dirichlet-to-Neumann map. 
Jussi Behrndt
math.SP