Medidas espectrais com singularidades em dinâmica quântica e unitária

Abstract

We present a study on the asymptotic behavior of some quantities in quantum and unitary dynamics. More specifically: (1) We prove sharp estimates on the time-average behavior of the squared absolute value of the Fourier transform of some absolutely continuous measures that may have power-law singularities, in the sense that their Radon-Nikodym derivatives diverge with a power-law order; we discuss an application to spectral measures of finite-rank perturbations of the discrete Laplacian. (2) We show that the power-law decay exponents in von Neumann’s Ergodic Theorem are the pointwise scaling exponents of a spectral measure at the spectral value 1. We also prove that, under an assumption of weak convergence, in the absence of a spectral gap, the convergence rates of the time- average in von Neumann’s Ergodic Theorem depend on sequences of time going to infinity.