Inverse Problems Of One-dimensional Schr?dinger Operator

One-dimensional Schr?dinger operator as a typical representative of differential operators,the research of this operator has a profound meaning to the further devel-opment of operator theory.It is well known that the research on the inverse problem is one of the fastest growing topic in the field of applied mathematics.They are originated from practical problems,such as the earth physics,the natural language processing,the quantum mechanics,the medical imaging and so on.The inverse problems of this operator have aroused high attention among the mathematicians and physicists for a long time and gotten many results.In this present paper,the inverse spectral and inverse scattering problems are studied first for the one-dimensional Schr?dinger operator.By using of the spectral data and/or scattering data,the uniqueness results about the potential and the boundary condition are proved and the reconstruction algorithms are provided.The mains works are as follows:Chapter one,we give the physical backgrounds of the one-dimensional Schr?dinger operators theory,and elaborate the research advances of the inverse spectral and inverse scattering problems,then introduce the main results of this paper.In the second chapter,the uniqueness problem of the regular one-dimensional Schr?dinger operator is discussed.Making use of the Marchenko's uniqueness theo-rem,it is shown that if the potential function is know on the partial interval,than the potential function and the boundary conditions can be uniquely determined on the finite interval in terms of choosing a set of appropriate eigenvalues from the infinite sets of spectra.In the third chapter,the finiteness for an inverse three spectra Schr?dinger problem is studied.Under condition that two boundary conditions at a fixed internal rational point a of(0,1)are different and known a priori,we show that there exist at most a finite number of differential operators corresponding to the three spectra of a one-dimensional Schr?dinger equation defined on[0,1],[0,a]and[a,1],respectively;and we give the accurate number of triplet.In the fourth chapter,we consider the inverse spectral problem for the sym-metric Schr?dinger operator with the known perturbance.We find this operator is PT-symmetric,this property show that a single Dirichlet spectrum suffices to determine the potential uniquely on the entire interval provided that the norms of both potential and kernel are sufficiently small.In the fifth chapter,the inverse spectral and scattering problem for the singular one-dimensional Schr?dinger operator are considered.Based on Gel'fand-Levitan and Marchenko integral equation,we show that if the partial knowledge of the potential given on a finite interval,the boundary condition and the potential are uniquely determined by the amplitude of the Jost function(the spectral function);and the potential is uniquely determined in terms of the phase of the Jost function(the scattering function),suppose that the boundary condition is known a priori.In particular,neither the bound state energies nor the norming constants(or the Marchenko norming constants)are needed to determine the potential uniquely.In the sixth chapter,we consider the inverse scattering problem for the singu-lar Schr?dinger operator with the boundary condition depending polynomials on a spectral parameter.The scattering data(scattering function,bound state energies and norming constants)properties of this operator,are investigated.We derive the modified Marchenko main equation of this operator and show that the potential is uniquely recovered and the reconstruction algorithm is provided in terms of the scattering data.