2018/03/19 by Anish Mallick, Mallick, Anish, M. Krishna +1
Mathematics · Materials Science · #Spectral Theory in Mathematical Physics #Random Matrices and Applications #Lanthanide and Transition Metal Complexes
paper · pdf · doi:10.48550/arxiv.1803.06895
In this paper, we consider Anderson type operators on a separable Hilbert space where the random perturbations are finite rank and the random variables have full support on ℝ. We show that spectral multiplicity has a uniform lower bound whenever the lower bound is given on a set of positive Lebesgue measure on the point spectrum away from the continuous one. We also show a deep connection between the multiplicity of pure point spectrum and local spectral statistics, in particular, we show that spectral multiplicity higher than one always gives non-Poisson local statistics in the framework of Minami theory. In particular, in higher rank Anderson models with pure-point spectrum, with the randomness having support equal to ℝ, there is a uniform lower bound on spectral multiplicity and in case this is larger than one the local statistics is not Poisson.