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 $L^2(\mathbb R^ν)$ with complex potential has absolute value at most a constant times $\|V\|_{γ+ν/2}^{(γ+ν/2)/γ}$ for $0<γ\leqν/2$ in dimension $ν\geq 2$. 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.

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