A Krein Space Approach to Elliptic Eigenvalue Problems with Indefinite Weights

where L is a symmetric elliptic operator and r is a locally integrable function on Rn. If r is of constant sign then this problem leads to a selfadjoint problem in the Hilbert space L2(|r|). In this note we are interested in the case when r takes both positive and negative values on sets of positive measure. Then the spectrum of the problem (1) is not necessarily real any more. Moreover it is not apparent that the spectrum does not cover the whole complex plane. Even when it is discrete there can be nonsimple eigenvalues. For ordinary differential equations the related completeness problem of the eigenfunctions has been studied extensively in recent years see for example [3, 6] and the references quoted therein. The corresponding question in the case of continuous spectrum has been addressed in [6].