Eigenvalue bounds for Schrödinger operators with complex potentials

We show that the absolute values of non-positive eigenvalues of Schrödinger operators with complex potentials can be bounded in terms of Lp-norms of the potential. This extends an inequality of Abramov, Aslanyan and Davies to higher dimensions and proves a conjecture by Laptev and Safronov. Our main ingredient are the uniform Sobolev inequalities of Kenig, Ruiz and Sogge.