We extend a result on dispersion for solutions of thelinear Schr\"odinger equation, proved by Firsova for operators withfinitely many energy bands only, to the case of smooth potentials in1D with infinitely many bands. The proof consists in an applicationof the method of stationary phase. Estimates for the phases,essentially the band functions, follow from work by Korotyaev. Mostof the paper is devoted to bounds for the Bloch functions. Forthese bounds we need a detailed analysis of the quasimomentumfunction and the uniformization of the inverse of thequasimomentum function.
Dispersion for Schrodinger Equation with Periodic Potential in 1D / Cuccagna, Scipio. - In: COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0360-5302. - STAMPA. - 33:11(2008), pp. 2064-2095. [10.1080/03605300802501582]
Dispersion for Schrodinger Equation with Periodic Potential in 1D
CUCCAGNA, Scipio
2008
Abstract
We extend a result on dispersion for solutions of thelinear Schr\"odinger equation, proved by Firsova for operators withfinitely many energy bands only, to the case of smooth potentials in1D with infinitely many bands. The proof consists in an applicationof the method of stationary phase. Estimates for the phases,essentially the band functions, follow from work by Korotyaev. Mostof the paper is devoted to bounds for the Bloch functions. Forthese bounds we need a detailed analysis of the quasimomentumfunction and the uniformization of the inverse of thequasimomentum function.Pubblicazioni consigliate
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