The propagation of the waves in periodic media at large time

Abstract

Asymptotic estimates at large time of the Green function to the wave equation with periodic coefficients are found. An estimate of the wave front is also obtained. It is shown that the spectral band (with number n=0,1,2,…) of the corresponding Hill operator ‘creates’ a wave having the front velocity c_n < 1 . Estimates of c_n in the terms of the gap lengths, the effective masses of the Hill operator are proved, both with fixed number n and their sums. Some extensions for more general cases (the Dirac operator with periodic coefficients, the Schrodinger operator with finite band potentials etc.) are also obtained.