Dispersion for Schrödinger equation with periodic potential in 1D. (English) Zbl 1163.35032

Author’s abstract: We extend a result on dispersion for solutions of the linear Schrödinger equation, proved by Firsova for operators with only finitely many energy bands, to the case of smooth potentials in 1D with infinitely many bands. The proof consists in an application of the method of stationary phase. Estimates for the phases, essentially the band functions, follow from work by Korotyaev. Most of the paper is devoted to bounds for the Bloch functions. For these bounds we need a detailed analysis of the quasimomentum function and the uniformization of the inverse of the quasimomentum function.


35Q55 NLS equations (nonlinear Schrödinger equations)
35B45 A priori estimates in context of PDEs
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