Principal Eigenvalues for Some Periodic-Parabolic Operators on N and Related Topics

We consider a periodic-parabolic eigenvalue problem with indefinite weight function m on N (N ≥ 3). Only under the assumption that the positive part of m has a certain decay we give necessary and sufficient conditions on m implying the existence of a unique positive principal eigenvalue. We prove that the zero solution of the corresponding parabolic equation looses stability at the principal eigenvalue. As a special case we obtain results on Schrodinger operators Δ + λm involving a coupling parameter λ, and on the stability properties of the corresponding Schrödinger semigroup et(Δ + λm).