Eigenvalue bounds for Schrödinger operators with complex potentials on compact manifolds

Abstract We prove eigenvalue bounds for Schrödinger operator - Δ g + V -₆+V on compact manifolds with complex potentials V. The bounds depend only on an L q L^{q} -norm of the potential, and they are shown to be optimal, in a certain sense, on the round sphere and more general Zoll manifolds. These bounds are natural analogues of Frank’s R. L. Frank, Eigenvalue bounds for Schrödinger operators with complex potentials, Bull. Lond. Math. Soc. 43 2011, 4, 745–750 results in the Euclidean case.