Inverse Spectral And Inverse Scattering Problems For Schr(?)dinger Operators

Schr(?)dinger operators appear in quantum mechanics,acoustics,chemistry,engineering mechanics,geophysics,electronics,meteorology and other fields of natural sciences,which have the wide applications.This thesis studies the inverse spectral and inverse scattering problems for the Schr(?)dinger operators,which consist in recovering the Schr(?)dinger operators from spectral data and/or scattering data.In the first chapter,we introduce the physical backgrounds and some applications of the Sclr6dinger operators,and summarize the current research advanees for the spectral-scattering theory and the main work and the novelties in this thesis.In the second chapter,a few notations are given,some knowledge related to the complex analysis is provided,and some properties of the solutions of Schr(?)dinger equations are introduced.In the third chapter,we investigate the Schr6dinger operator on a finite interval[0,1]with discontinuous conditions at 1/2.Using the Hadamard factorization theorem and the Phragmen-Lindel6f theorem,we show that if the potential is known a priori on a subinterval[b,1]with b≥1/2,then parts of two spectra(or N spectra,N≥2)can uniquely determine the potential and all parameters in discontinuous conditions and boundary conditions.For the case b<1/2,parts of one spectrum can uniquely determine the potential and a part of parameters.In the fourth chapter,we study the inverse eigenvalue problem for the Schr(?)dinger operator on the halfline with the Robin boundary condition.It is shown that a particular set of eigenvalues can uniquely determine the potential.Moreover,the reconstruction algorithm for recovering the potential is provided by using the method of power series analytic continuation and the Gelfand-Levitan equation.In the fifth chapter,the inverse resonance problems for the Schr(?)dinger operators with the potential compactly supported in[0,1]are considered.For the half line case,we prove that if the potential is known a prior on a subinterval of the whole interval,then only a part of eigenvalues and resonances can uniquely determine the potential,Moreover,the relationship between the proportion of the needed data and the length of the subinterval on the given potential is revealed.For the full line case,(1)It is shown that if the potential is known a priori on[0,1/2],then the unique recovery of the potential on the whole interval from all eigenvalues and resonances is valid;(2)If the potential is known a priori on[0,a]with a>1/2,then infinitely many eigenvalues and resonances can be missing for the unique determination of the potential;(3)If the potential is known a priori on[0,a]with a<1/2,then all eigenvalues and resonances plus a part of so-called sign-set can uniquely determine the potential;(4)It is also shown that all eigenvalues and resonances,together with a set of logarithmic derivative values of eigenfunctions and wave-functions at 1/2,can uniquely determine the potential.In the last chapter,the matrix Schr6dinger operators are studied.For the half line case,it is shown that the scattering data,which consists of the scattering matrix and the bound state data,uniquely determines the self-adjoint potential and the unitary matrix in the boundary condition.Moreover,it is also shown that only the scattering matrix uniquely determines the self-adjoint potential and the boundary condition if either the potential exponentially decreases fast enough or the potential is known a priori on(a,+∞)(a>0).For the full line case,it is shown that the left(or right)reflection coefficient uniquely determine the self-adjoint potential if either the potential exponentially decreases fast enough or the potential is known a priori on(-∞,b)(or(b,+∞)).We also study the inverse scattering problems on a noncompact star graph.For the graph containing only one finite edge,we give the uniqueness theorems and reconstruction algorithms for missing partial bound state data;for the graph containing more than one finite edges,we give the uniqueness theorems and reconstruction algorithms for determining the potentials on the whole graph.