Random Schr\"odinger operators with complex decaying potentials

TL;DR

This paper establishes probabilistic bounds on eigenvalues of random Schr"odinger operators with complex decaying potentials, effectively doubling the exponent range compared to deterministic bounds, using advanced probabilistic and analytical techniques.

Contribution

It demonstrates that eigenvalues of random Schr"odinger operators with complex decaying potentials can be bounded in terms of an $L^q$ norm with a doubled exponent range, improving upon previous deterministic results.

Findings

Eigenvalues are bounded with high probability in terms of $L^q$ norm.

The exponent $q$ can be doubled in the random setting.

The approach uses ideas related to almost sure scattering for lattice Schr"odinger operators.

Abstract

We prove that the eigenvalues of a continuum random Schr\"odinger 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 $q\leq d+1$. 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.