Eigenvalue bounds for Schrödinger operators with complex potentials. II

Abstract

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