Mathematical Research Letters

Volume 27 (2020)

Number 3

Exact dynamical decay rate for the almost Mathieu operator

Pages: 789 – 808



Svetlana Jitomirskaya (Department of Mathematics, University of California at Irvine)

Helge Krüger (Department of Mathematics, California Institute of Technology (Caltech), Pasadena, Cal., U.S.A.)

Wencai Liu (Department of Mathematics, University of California at Irvine; and Department of Mathematics, Texas A&M University, College Station, Tx., U.S.A.)


We prove that the exponential decay rate in expectation is well defined and is equal to the Lyapunov exponent, for supercritical almost Mathieu operators with Diophantine frequencies.

S.J. was supported by NSF DMS-1401204 and DMS-1901462. W. L. was supported by NSF DMS-1700314/2015683, DMS-2000345, the AMS-Simons Travel Grant 2016-2018 and the Southeastern Conference (SEC) Faculty Travel Grant 2020-2021. S.J. and W.L. are grateful to the Isaac Newton Institute for Mathematical Sciences, Cambridge, for its hospitality, supported by EPSRC Grant Number EP/K032208/1, during the 2015 programme Periodic and Ergodic Spectral Problems where an important progress on this work was made.

Received 6 December 2018

Accepted 8 June 2019

Published 20 August 2020