Random Schr\"odinger operators with complex decaying potentials

We prove that the eigenvalues of a continuum random Schr"odinger operator $-\mathrm{\Delta }+V_{\omega }$ of Anderson type, with complex decaying potential, can be bounded (with high probability) in terms of an $L^q$ norm of the potential for all $q\le d+1$. This shows that in the random setting, the exponent $q$ can be essentially doubled compared to the deterministic bounds of Frank (Bull. Lond. Math. Soc., 2011). This improvement is based on ideas of Bourgain (Discrete Contin. Dyn. Syst., 2002) related to almost sure scattering for lattice Schr"odinger operators.