Eigenvalue bounds for Schrödinger operators with complex potentials. II

Laptev and Safronov conjectured that any non-positive eigenvalue of a Schrödinger operator -Δ+V in L^2 (R^ν) with complex potential has absolute value at most a constant times ||V||^(γ+ν/2)/γ)_(γ+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 < ν/2. In addition, we prove various bounds that hold, in particular, for positive eigenvalues.