Spherically Averaged Endpoint Strichartz Estimates For The Two-dimensional Schr(?)dinger Equation With A Potential

Since the 1920s, theory of Schro¨dinger equations has constantly been a central topicof the subject–Mathematical Physics. This theory has cover many aspects, such as spectralanalysis, scattering properties, and Lp ?Lq estimates, local smoothing estimates, weightedestimates, maximal operator's estimates, Strichartz time-space estimates for the solutions.The study on Schro¨dinger equations is of profound theoretical in mathematics. It is also ofpractical significance because of its physical background. The Strichartz estimates studiedby this thesis, is an embranchment of Schro¨dinger equations, which was put forward byStrichartz firstly at 1977 and keep attracting mathematician's attention during the past sev-eral decades. Until now, mathematicians have got fruitful achievements about the study onStrichartz estimates for Schro¨dinger equations, such as endpoint Strichartz estimates for thehigh-dimensional Schro¨dinger equations, spherically averaged endpoint estimates for thetwo-dimensional equations, and achievements on estimates about kinds of potentials. How-ever, there are still many problems in this field remain unsolved. The purpose of this thesis,is to prove the endpoint Strichartz estimates for the two-dimensional Schro¨dinger equationwith a potential by spherically averaged method.The whole thesis will be separated into three chapters. The fist chapter is an introduc-tion to Quantum mechanics and the theory of Schro¨dinger equations, the second chapteris a summation of the theory of Strichartz estimates, in the third chapter, the main resultwill be stated and proofed. More specifically, the proof of the main result will base on Taoand D'Ancona's achievements, and be finished by decomposing CIL2 in a direct sum andconstructing a compress mapping on every subspace. Finally, the conclusion of this thesiswill be summarized and prospected.