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 $Ω$ are investigated. It is shown that under an abstract regularity assumption the nonreal spectrum of the associated elliptic operator in $L^2(Ω)$ is bounded. In the special case that $Ω=R^n$decomposes into subdomains $Ω_+$ and $Ω_-$ with smooth compact boundaries and the weight function is positive on $Ω_+$ and negative on $Ω_-$, it turns out that the nonreal spectrum consists only of normal eigenvalues which can be characterized with a Dirichlet-to-Neumann map.

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