Spectral theory of elliptic differential operators with indefinite weights

Abstract

The spectral properties of a class of non-self-adjoint second-order elliptic operators with indefinite weight functions on unbounded domains Ω are investigated. It is shown, under an abstract regularity assumption, that the non-real spectrum of the associated elliptic operators in L²(Ω) is bounded. In the special case where Ω = ℝⁿ 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 non-real spectrum consists only of normal eigenvalues that can be characterized with a Dirichlet-to-Neumann map.