Research On The Inverse Spectral Problems Based On Krein-Nudelman Interpolation

As the essential mathematical tool and theoretical basis in the fields of the quantum mechanics,the earth physics and other engineering and techniques,the theory of differential operators has attracted great interests of the domestic and overseas scholars and aroused their high attention.Up to now,it has become one of the research fields that are developed fastest and most active in the mathematical physics.The research on the Krein strings is one of the important subjects in the theory of differential operators.This dissertation mainly focus on the research on the direct and inverse spectral problems of the Krein strings in two different cases which are the Stieltjes strings and the operators generated by the Schrodinger equations.In this dissertation,we take Krein-Nudelman interpolation theory as the main tool,we study the inverse problems of the Stieltjes strings and Schrodinger operators with the mixed given spectral data.Here the Stieltjes strings are generated by the Krein strings with the mass distribution function being the step function and the Schrodinger equations are generated by the Krein strings with the mass distribution function being sufficiently smooth.We provide the sufficient and necessary conditions to the corresponding inverse problems.More precisely,in this dissertation for the Stieltjes strings which are damped at one of the ends and at an interior point,respectively,we uniquely determine the mass distribution and propose the algorithms of recovering the mass distribution on the strings.We prove the uniqueness of the potential of the Schrodinger operators on the half line and show the reconstruction algorithm of the potential.It should be noticed that the inverse problems mentioned above can convert to the Krein-Nudelman's interpolation problems.Consequently,we solve the inverse problems in terms of the Krein-Nudelman's interpolation formulae and the modified Krein-Nudelman's interpolation formulae.The main works of this dissertation are given as follows:We study the inverse problems for a Stieltjes string damped at one of the ends with mixed spectral data.In this case,the eigenvalues are all suited in the open upper half-plane C+.In virtue of the appropriate transformation,we convert the given spectral data and a part of the mass distribution on the string to the relative Krein-Nudelman's interpolation problem.By solving the problem,we establish the relation between one of its solutions and the Stieltjes string.Consequently,we uniquely determine the unknown mass distribution on the Stieltjes string and furthermore provide the algorithm by the Stieltjes continued fraction expansion.We study the inverse problems for a Stieltjes string damped at an interior point with mixed spectral data.Note that this Stieltjes string can be regarded as a simple star graph constructed by two strings and the real eigenvalues may occur for this Stieltjes string.Analogously to the Stieltjes string damped at one of the ends,we convert the mixed given spectral data to the related Krein-Nudelman's interpolation problem.We solve the inverse problem by the connection between the real eigenvalues and the relative S-function and then uniquely determine the mass distribution on this string.Moreover,for the case where the Stieltjes string has only complex eigenvalues,we propose the solution to the inverse problem and provide the algorithm of recovering the unknown mass distribution on the string.We study the direct and inverse spectral problems for a class of star-shaped graphs of Stieltjes strings.We do the research on the star graph which damped at a pendant vertex.According to the traditional approach in the spectral theory and complex analysis,we study the spectral problems of the star graph.We analyze the distribution of the eigenvalues and show the continued fraction expansion of an S-function associated with the graph.Furthermore,we solve the inverse problems of the existence,the uniqueness and the reconstruction for the star graph.We study the inverse resonant problems of a Schrodinger operator on the half line.If the real and continuous potential is compact supported,then the operator has the resonances with the asymptotic expansion in the open lower half plane C-.We convert the mixed given data to the modified Krein-Nudelman's interpolation problem and show the solution by the modified interpolation formulae to obtain an S0 function closely related to the potential.By analyzing the relation between the densities of the zeros and poles of the S0-function and the density of the resonances,we uniquely determine the potential.Furthermore,we establish and solve the Marchenko integral equation to recover the unknown potential.