Random Schrödinger operators with complex decaying potentials

Abstract

We prove that the eigenvalues of a continuum random Schrödinger operator $-\Delta +V_{\omega }$ 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 Schrödinger operators.

Minor corrections. Comments welcome

References in corpus (2)

• Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials
• Weak coupling limit for Schrödinger-type operators with degenerate kinetic energy for a large class of potentials