Inverse Spectral Problems For Radial Schr(?)dinger Operators

The radial Schrodinger operator originates from the quantum mechanics and is derived from the stationary Schrodinger equation with spherically symmetric potential through the separation of variables.This kind of operators are widely used in the fields such as acoustics,optics,radio physics,fluid dynamics,atomic and nuclear physics,etc.This thesis is mainly focused on the inverse spectral problems of the radial Schrodinger operators,which are recovered from the spectrum data.In quantum mechanics,the physical meaning of this problem is to recover the potential field through the energy of the particle and other relevant observable data.The results of the thesis enrich the spectral theory of the radial Schrodinger operators and provide the theoretical basis for the numerical calculation of restoring the radial Schrodinger operators.The contents of the thesis are as follows.In chapter 1,we introduce the physical background of the radial Schrodinger operators,and summarize their current research progress.Then we present main results of the thesis.In chapter 2,we construct a kind of singular transformation operators for the radial Schr(?)dinger operators,and by using the singular transformation operators,we give a criterion for the uniqueness of the inverse spectral problems.That is,we construct a function system by using the given eigenvalues,and obtain the uniqueness result of the inverse problems by using the closedness condition of the function system.Since the closedness condition is general,the uniqueness result contains some classical results of the inverse problems for the radial Schrodinger operators.In addition,under a hypothesis of an isomorphic condition about a linear operator,we prove that the closedness condition is not only sufficient but also necessary for the inverse problems.In chapter 3,in view of the discontinuity and singular potential(for example,Coulomb potential)that may occur in practical problems,we consider the inverse spectral problem for radial Schrodinger operators with discontinuous conditions.This problem occurs in the physical model when the propagation medium has faults,thus the corresponding boundary value problem contains discontinuous conditions.Applying techniques of complex analysis such as Hadamard’s product theorem,we give two uniqueness theorems of the inverse spectral problems.In chapter 4,for the radial Schr(?)dinger operators on a finite interval,if the eigenvalues of two operators corresponding to two potentials are the same under two different boundary conditions,then the two potentials are equal.However,when the above eigenvalues are not exactly the same,how about the relationship between the two potentials?By constructing an auxiliary operator and using the Mittag-Leffler expansion,we give an accurate representation of the difference between the two potentials according to the given eigenvalues and the related solutions of the equations.In chapter 5,applying results of the inverse spectral problems for the radial Schr(?)dinger operators,we study the inverse resonance scattering problem.Using the eigenvalues and resonances corresponding to a fixed angle momentum quantum number of the scattering problem as the priori data,by Hadamard’s product theorem,we give a uniqueness theorem for the inverse resonance scattering problem.In chapter 6,applying results of the inverse spectral problems for the radial Schr(?)dinger operators,we study the inverse interior transmission eigenvalue problem.Using the eigenvalues corresponding to a fixed angular momentum quantum number of the transmission eigenvalue problem as the spectral data,by some tools of complex analysis,we provide some uniqueness theorems for the inverse transmission eigenvalue problem.