Eigenvalue bounds for Schrödinger operators with complex potentials on compact manifolds

We prove eigenvalue bounds for Schrödinger operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>g</m:mi> </m:msub> </m:mrow> <m:mo>+</m:mo> <m:mi>V</m:mi> </m:mrow> </m:math> {-Δg+V} on compact manifolds with complex potentials V . The bounds depend only on an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi>q</m:mi> </m:msup> </m:math> {Lq} -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.