New $L^p$ Estimates for Solutions to the Schrödinger Equations and Time Asymptotic Behavior of Observables

Abstract

IntroductionWe consider the Schrodinger operator H=H Q +V in the Hilbert space L\R n }, 72^1, where H 0 is the self -ad joint realization of -A in L\R n } and V is a symmetric operator with //o-bound less than one. This paper is mainly devoted to obtaining detailed informations about the asymptotic behavior in time of the following quantities:where (j)^L\R n ) is an appropriate initial datum. We obtained some new estimates for (l)-(3).For a nice initial datum 0, it is reasonable to expect that e~i tH^^Lp (R n ) for all t^R and that the map R^t^e~i tH^^Lp (R n ) is continuous even when p^2. By taking the asymptotic behavior of the Schrodinger free evolution group into account, it is natural to think that for scattering solutions (i.e., e~U H (f> with 0 orthogonal to any eigenvector of H\ (l)-(3) decay as £->±oo provided 2<p<°o. The local L 2 -decay of scattering solutions has been extensively studied by many mathematicians (see [14] The contents of this paper are as follows. In § 1 we give some preliminary lemmas. In §2 we study the Schrodinger evolution group in the weighted Sobolev spaces. In § 3 we state the pseudo-conformal conservation law for the Schrodinger equation. The pseudo-conformal conservation law for some class of non-linear Schrodinger equations was first observed by Ginibre & Velo [8]. Then, the non-linear theories have been developed by many authors (see [9] [10][11] [37]). In §4 we establish the general existence theory of //-solutions to the Schrodinger equations. Various //-estimates will be obtained. We use these estimates in § 5 and § 6. We investigate scattering solutions in § 5. Sufficient conditions for the quantities (l)-(3) to decay in time will be given. The purpose of §6 is to determine the rate of decay of (l)-(3) for short-range potentials and repulsive potentials. Acknowledgements