Eigenvalue bounds for Schrödinger operators with complex potentials. II

Laptev and Safronov conjectured that any non-positive eigenvalue of a Schrödinger operator −Δ+Vin L2(Rν) with complex potential has absolute value at most a constant times ∥V∥(γ+ν/2)/γγ+ν/2 for 0<γ≤ν/2 in dimension ν≥2. We prove this conjecture for radial potentials if 0<γ<ν/2 and we 'almost disprove' it for general potentials if 1/2<γ