2013/12/18 by M.A. Astaburuaga, Astaburuaga, M. A., Olivier Bourget +3
Mathematics · Physics and Astronomy · Computer Science · #Spectral Theory in Mathematical Physics #Quantum chaos and dynamical systems #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.1312.5299
Let U be a unitary operator defined on some infinite-dimensional complex Hilbert space \cal H. Under some suitable regularity assumptions, it is known that a local positive commutation relation between U and an auxiliary self-adjoint operator A defined on \cal H allows to prove that the spectrum of U has no singular continuous spectrum and a finite point spectrum, at least locally. We prove that under stronger regularity hypotheses, the local regularity properties of the spectral measure of U are improved, leading to a better control of the decay of the correlation functions. As shown in the applications, these results may be applied to the study of periodic time-dependent quantum systems, classical dynamical systems and spectral problems related to the theory of orthogonal polynomials on the unit circle.