Inverse Problems For Two Kinds Of Differential Operators With Discontinuities

The boundary value problems with discontinuities arise in many aspects,such as mathematics,physics,electronics and geophysics,which has practical applications in the recognition of material feature for electronic devices and the determination of the density for the earth.The spectral characteristics and the inverse problems for differential operators with discontinuous points inside the interval are studied in thesis,which consists in the characterization of spectral features,uniqueness theorems and reconstruction algorithms.In Chapter 1 we review the current progress of the research on Sturm-Liouville operators and Dirac operators,as well as the main works in this thesis.In Chapter 2 we study the half inverse problem and inverse nodal problem for the impulsive Sturm-Liouville operator on the interval(0,π)with discontinuity.For the half inverse problem,the spectral characteristics are described,and using Hadamard’s factorization theorem and Phragmén-Lindel(?)f’s theorem,we show that if the potential is known a priori on(0,1+α/4π)(the impulsive parameter α ∈(0,1)),then only one spectrum and parameter in one boundary condition can uniquely determine the potential on the whole interval.If the potential is known on(1+α/4π,π),then only one spectrum and parameter in another boundary condition can uniquely determine the potential on the whole interval.A set of values of the logarithmic derivative of eigenfunctions at the internal point(1+α)π/4)and one spectrum are taken as data to proved the interior inverse problem by the known spectral information.For the inverse nodal problem,we give the asymptotic estimates of nodal points for the eigenfunctions and also prove the uniqueness theorem using the node data.In Chapter 3 we concern with some results for Sturm-Liouville operator on the interval(0,π)with discontinuity,including the spectral characteristics,the solvability of the inverse problem and the inverse resonance problems.We obtain the alternation for eigenvalues and the oscillation for eigenfunctions,prove the uniqueness theorem and provide the necessary and sufficient conditions for the solvability of the inverse problem.For the inverse resonance problems of this operator,combining the relationship between Cauchy data {Kx(7r,t),Kt(π,t)} and the spectrum,we get that one spectrum can uniquely determine the potential on the whole interval.By conversing the problem to Regge problem and using Riesz bases,we prove that one spectrum can uniquely determine the potential when the discontinuous point d=π/2 or 0<d<π/2 and present the reconstruction algorithms for the inverse resonance problem.In Chapter 4 we study the Sturm-Liouville operator on the interval(0,π)with discontinuities and the periodic boundary condition,which consists in the spectral characteristics,the uniqueness theorem and the reconstruction algorithms.When the number of discontinuous points is one or two,we give the spectral properties,respectively and prove that two spectra and the corresponding set of symbol can determine the potential.We also obtain the corresponding reconstruction algorithm.In Chapter 5 the half inverse problem and the uniqueness theorem for the Dirac operator on the interval(0,π)with discontinuity are considered.We show the half inverse problem,that is,if the potential is known on(π/2,π)and all parameters in discontinuous conditions are given,then only one spectrum can uniquely determine the potential on the whole interval,and using Hadamard’s factorization theorem,we present the reconstruction algorithm.When potential and parameters are real,we present the spectral characteristics and prove the uniqueness theorems by using supplementary new data.When potential and parameters are complex,we prove the uniqueness theorem for this operator by Weyl function or two spectra.Finally,in Chapter 6 we discuss some unsolved problems for the Sturm-Liouville operators and Dirac operators with discontinuities.