Studies On Solving The Direct And Inverse Problems For A Class Of Schr(?)dinger Equations

The Schr(?)dinger equation is a basic equation in non-relativistic quantum mechanics.It describes the law of motion followed by the "material wave" in a quantum system.The Schr(?)dinger equation can deal with non-relativistic problems,which is widely used in the fields of atom,molecule,quantum physics,and chemistry.In this paper,we mainly study the numerical theory and computational method of direct and inverse problems for the steady-state Schr(?)dinger equation in unbounded domain.The inversion results of potential function are obtained by combining theoretical analysis with numerical experiments.Firstly,the direct problem of the steady-state Schr(?)dinger equation is studied in unbounded domain.Using PML method,the unbounded domain problems are transformed into the boundary value problems of partial differential equations in bounded computational domain.The existence and uniqueness of solutions to the variational problems for the boundary value problems of partial differential equations are proved.The PML equation is solved numerically by the finite difference method,and the numerical results of the direct problem are obtained.The effects of different parameters on the numerical results are analyzed by numerical experiments.Secondly,the potential function of the Schr(?)dinger equation is inverted by the primal-dual method.Considering the ill-posedness of the inverse problem,we use the generalized cross-validation method to select the regularization parameters,thus giving a new inversion method.Based on multi-frequency measurement and initial guess selection strategy,a multi-frequency iterative algorithm from low frequency to high frequency is presented.The effectiveness of the proposed method is verified by the numerical experiments.