Time almost periodicity for solutions of Toda lattice equation with almost periodic initial datum

This paper analyzes the initial value problem for the Toda lattice with almost periodic initial data: let $J(t; J_{0})$ denote the family of Jacobi matrices which are solutions of the Toda flow equation with initial condition $J(0; J_{0})=J_{0},$ then, the given almost periodic datum $J_{0}$ is a discrete linear Schr & ouml;dinger operator with almost periodic potential, which plays a fundamental role in our considerations. We show that, under some given hypotheses, the spectrum of the Schr & ouml;dinger operator is pure absolute continuous and homogeneous (measure-theoretically) by establishing exponential asymptotics on the size of spectral gaps. These two conclusions enable us to show the boundedness and almost periodicity in the time of solutions for Toda lattice equation with almost periodic initial data. As a consequence, our result presents a positive answer to the discrete Deift's conjecture [Some open problems in random matrix theory and the theory of integrable systems. Integrable Systems and Random Matrices (Contemporary Mathematics, 458). American Mathematical Society, Providence, RI, 2008, pp. 419-430; Some open problems in random matrix theory and the theory of integrable systems. II. SIGMA Symmetry Integrability Geom. Methods Appl. 13 (2017), Paper no. 016].