Uniqueness And Continuous Dependence In The Inverse Problem Of Schr?dinger Equation

This article studies the uniqueness of the potential term in the inverse problem of Schrodinger equation.The equations studied are as follows:(?) Given the behavior of the solution in the internal region U(?)?,not only the uniqueness of the potential term p(x)is obtained,but also the continuous dependence on the potential term is obtained.In terms of research methods,this paper mainly proves the uniqueness of the potential term by combining the results of the internal Carleman inequality after even continuation of the equation,and using the mid-point moment control.First,the conclusion of the global Carleman estimation of the Schrodinger operator under the non-degenerate weight is given,and the proof derivation process is given in detail.Then the core content of this paper is to study the uniqueness of the potential p(x)in the inverse problem of Schrodinger equation.The main idea is to transform the nonlinear inverse problem into a linear case by making a difference between the two equations,and then use the nature of the inverse solution of the Schrodinger equation to evenly extend the solution to make the time range in the equation from(0,T)to(-T,T),then use the value of t=0 at the middle moment to construct the equation,and finally combining Carleman's inequality to obtain the internal upper solution control of the potential term p(x).At the end of this paper,the results of the inequality of internal control potential are generalized to bilateral estimates,and the continuous dependence of the solution on the potential in the internal region is obtained.First,based on the principle of homogeneity,bilateral estimates are obtained,and then the energy of the corresponding Schrodinger equation is generalized.Observe the inequality,and then use the compactness-uniqueness conclusion to obtain a symmetrical bilateral estimate.