Random Schrödinger operators with breather potentials as a paradigmatic model for non-linear influence of randomness

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In the project we study spectral, dynamical, and statistical properties of Schroedinger operators with random breather potential.Such models feature a non linear dependence on a canonical sequence of independent, identically distributed random variables.This poses an additional challenge for the mathematical analysis and physical understanding compared to models with linear random parameters as, for instance the alloy type Schroedinger operator, or its discrete lattice cousin, the Anderson model.A driving question for our research is the persistence of typical signatures of localization (in appropriate disorder/energy regimes) for such non linear models, compared to linear ones. In particular, we want to pursue the question whether sufficient disorder leads to localization, independently of whether the considered model is linear or not, and thus clarify one aspect of universality of the phenomenon of Anderson localization. As a benefit of our analysis we expect not only a better understanding of the physical mechanism of localization, but identification and development of novel mathematical tools crucial for the analysis of random differential operators. We choose to study random breather Schroedinger operators because this class is on one hand tangible enough to be amenable to explicit calculations, and includes {solvable models, on the other hand it has paradigmatic properties common to many random potentials with non linear parameter influence: The analysis of volume and geometric structure of level sets of (different configurations of) random potentials is at the core of understanding intricate spectral properties.Breather models have the additional trait that they come in different varieties of difficulty: pointwise monotone potentials, on average monotone potentials, and truly non monotone ones. We intend to establish Lifshitz tails, initial scale estimates, Wegner estimates, multi scale analysis/proofs of localization, (de)correlation estimates, as well as spectral statistics for the random Schroedinger operators considered. Our group has already experience with the analysis of random Schroedinger operators, both with linear parameter dependence, e.g. sign changing alloy type models, as well as with non linear parameter dependence, e.g. random quantum waveguides.