Dispersive and Strichartz estimates for Schrodinger equations

The present paper is concerned with LP -smoothing properties of solutions to time dependentSchriidinger equations. Vve study two kinds of V-smoothing properties, namely theStrichar:f;z estimates and the dispersive estimates.The first part is concerned with Strichartz estimates for Schriidinger equations onscattering manifolds which are non-compact manifolds with asymptotically conic ends.It is shown that time-local Strichartz estimates, outside a large compact set centered atorigin, hold for any admissible pair (including the endpoint). Moreover, we prove globalin space Strichartz estimates under the nontrapping condition of the geodesic flow. \¥ealso study a relationship between Strichartz estimates and microlocal properties of thesolution. More precisely, it is shown that (local in time) Strichartz estimates follow fromthe dispersive estimates for spatial and frequency localized solutions. This part is basedon a paper " Strichartz estimates for Schriidinger equations on scattering manifolds " .The second part is concerned with dispersive estimates for scattering solutions toone-dimensional Schriidinger equations with potentials. We prove a weighted dispersiveestimate for the non-resonant state with stronger time decay than the case of the usualunweighted estimate. Furthermore asymptotic expansions in time of scattering solutionsare given. This part is based on a paper "Dispersive estimates and asymptotic expansionsfor Schriidinger equations in dimension one" to appear in Journal of the MathematicalSociety ofJapan.報告番号: 甲27194 ; 学位授与年月日: 2011-03-24 ; 学位の種別: 課程博士 ; 学位の種類: 博士(数理科学) ; 学位記番号: 博数理第375号 ; 研究科・専攻: 数理科学研究科数理科学専