Eigenvalue bounds for Schr\"odinger operators with complex potentials. II

Laptev and Safronov conjectured that any non-positive eigenvalue of a Schr"odinger operator $-\mathrm{\Delta }+V$ in $L^2({\mathbb{R}}^{\nu })$ with complex potential has absolute value at most a constant times $\mathrm{\parallel }V{\mathrm{\parallel }}_{\gamma +\nu \mathrm{/}2}^{(\gamma +\nu \mathrm{/}2)\mathrm{/}\gamma }$ for $0<\gamma \le \nu \mathrm{/}2$ in dimension $\nu \ge 2$. We prove this conjecture for radial potentials if $0<\gamma <\nu \mathrm{/}2$ and we `almost disprove’ it for general potentials if $1\mathrm{/}2<\gamma <\nu \mathrm{/}2$. In addition, we prove various bounds that hold, in particular, for positive eigenvalues.