Scattering For Fractional Schr(?)dinger Operators With Short Range Potentials

Scattering is an effective method to study microscopic particles in nature. There arealso many scattering phenomena in nature, such as light scattering. For the quantum scatter-ing, a number of research results have been obtained. Classical scattering theory is dividedinto the cases of short range and long range potentials. If wave operators with potentialsexist, such potentials are called short range potentials. If modified wave operators with po-tentials exist, such potentials are called long range potentials. The case with short rangepotentials is relatively mature, and the case with long range potentials is not perfect. Com-pared with classical scattering theory, the integral order of high order Schr(o|¨)dinger operatorscattering theory is not perfect. When main operators are linear partial differential opera-tors and potentials V are variable coefficient partial differential operators, H(o|¨)rmander fullystudied the case with short range potentials and gave some introductions for the case of longrange. Recently, scholars start to pay attention to fractional Schr(o|¨)dinger operator. For theoperator (-â–³)^α/2, physically fractional Schr(o|¨)dinger equation usually is considered in thesituation of 1 <α≤2. Mathematically, Schr(o|¨)dinger equation is considered in a widerrange ofα> 0. At present, there are some results about fractional Schr(o|¨)dinger operators,such as smoothing estimates, intrinsic ultracontractivity at the aspect of semigroups, the ex-istence and uniqueness of nonlinear Schr(o|¨)dinger equations'solutions. And we focused onthe scattering problem of fractional Schr(o|¨)dinger operators.The scattering problem we study is mainly for these two systems, (-â–³)^α/2 and(-â–³)^α/2 + V . The main arrangements are as follows: we introduce the basic conceptsand major work, then we study the resolvent R₀(z) of the operator (-â–³)^α/2. For someappropriate V , V R₀(z) is a compact operator in the appropriate space. Using such potentialV , we define the concept of the short range perturbation. Finally, using the stationaryphase method and the compactness of the operator V which is similar to the method ofH(o|¨)rmander, we prove the existence of wave operators in short range perturbations. After thecompletion of the main proof, we give two asymptotic behavior of solutions of fractionalSchr(o|¨)dinger equation's solutions. Then, we point some areas that can be improved.