Localization for the random displacement model
The random displacement model is a random Schrödinger operator in the continuum with a random perturbation of the periodic potential. In this article, the authors prove spectral localization (a.s. pure point spectrum and exponentially decaying eigenfunctions) and dynamical localization in an interval above the ground state energy. A key ingredient is previous work by two of the authors characterizing the configurations of random displacements that minimize the ground state energy. This is turn is derived from a careful analysis of the ground state energy of Schrödinger operators in a box with von Neumann boundary conditions, which shows that the potential that minimizes the ground state energy is centered in a corner. The structure of the localization proof is standard: (i) smallness of the integrated density of states and (ii) Wegner estimate, used as inputs to the multiscale method. While (i) \& (ii) are proved in details in the paper and both use the a priori knowledge on the ground state energy, the running of the multiscale method from them is known and only briefly discussed for pedagogical reasons.
scientific article; zbMATH DE number 6024992
scientific article