The Scattering Problem Of Fractional Schr(?)dinger Operators With Critical Potential

This paper mainly studies the scattering problem of the fractional Schr（?）dinger operator and related spectrum analysis.We first propose a kind of critical short range potential for the fractional Schr（?）dinger operator,and then prove the existence of the wave operators,then perform a detailed spectral analysis of the fractional Schr（?）dinger operator of this kind of potential and prove the asymptotic completeness of the wave operators.Finally,we consider the scattering problem of the higher order Schr（?）dinger operators under another kind of short range potential.This article consists of five chapters in total.The first chapter introduces the physical background of Schr（?）dinger operator,the development history and general research field of Schr（?）dinger operator,and the gen-eral research methods and development history for scattering theory of Schr（?）dinger operator,as well as spectrum analysis related to scattering theory,and finally we give the main research content of this article.The second chapter considers the scattering theory of the fractional Schr（?）dinger operator（-Δ）^s/2+V（x）.For the scattering theory of the Schr（?）dinger operator,people usually only study the second order,that is,the general physical background,and the results of the high order or fractional cases are not many and incomplete.The reason is that the traditional scattering theory research methods are too demanding.Our method is through the definition of potential V（x）is given on a suitable L²dense subspace,its sufficient and necessary conditions are established,and specific potential examples are given.Finally,the Cook’s method is used to directly calculate the existence of the wave operators of the fractional Schr（?）dinger operator.Chapter 3 continues to study the scattering theory of the fractional Schr（?）dinger operator（-Δ）^s/2+V（x）.After we have obtained the existence of its wave operator,under the premise,a more detailed analysis is performed on the spectrum of the frac-tional Schr（?）dinger operator under the potential condition,and finally the asymptotic completeness of the wave operator is obtained.This kind of problem is in the second order,higher order the situation has been studied in detail,and in the fractional case,the”good”result is often not obtained due to the insufficiency of the method.We first use a method similar to H（?）rmander to establish a free resolvent estimate and obtain the weighted version of the free resolvent uniform estimation,and then using the particularity of the potential which we defined,we get the eigenvalues and related properties of the eigenfunction of the fractional Schr（?）dinger operator（-Δ）^s/2+V（x）.On this basis,we use the”distorted”Fourier transform to obtain the asymptotic com-pleteness of the wave operators.Since we are concerned about（-Δ）^s/2,the special explanation of the characteristic equation,the eigenvalues and the relevant properties of the eigenfunction obtained above are only partial,and finally we show that this is also the whole.The fourth chapter considers the scattering theory of the high order Schr（?）dinger operator（-Δ）^m+V（x,D）.Based on the Kato smoothing theory method,we fully prove the Agmon-Kato-Kuroda theorem.First,we use the Stein-Tomas restriction theorem to establish a suitable L²dense subspace.We notice the correlation between the second order case and the higher order case,and the particularity of the second order case gives the free resolvent formula and weighted free resolvent estimation of the high order case,and then we use the Kato smoothing theory to give the defini-tion of potential,and give specific potential examples.For our final results,we give some lemma of the potentials and establishes the principle of limit absorption,and finally proves the existence and asymptotic completeness of wave operators and related spectral properties.Chapter 5 gives further thoughts on the previous three aspects.