Spectral theory of elliptic differential operators with indefinite weights

The spectral properties of a class of non-self-adjoint second-order elliptic operators with indefinite weight functions on unbounded domains Omega are investigated. It is shown, under an abstract regularity assumption, that the non-real spectrum of the associated elliptic operators in L-2(Omega) is bounded. In the special case where 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 non-real spectrum consists only of normal eigenvalues that can be characterized with a Dirichlet-to-Neumann map.