Research On The Dependence Of Eigenvalues And Inverse Spectral Problems For Certain Differential Operators

In mathematics and physics,differential operators are important tools for studying changes,motion,and the relationship between function spaces and operator spaces.They are of great significance for understanding natural phenomena,solving practical problems,and promoting the development of mathematical theory.They are also widely used in other disciplines.With the emergence of numerous problems in application fields,some physical problems are transformed into differential operator problems with internal discontinuity.For example,the vibration problem of the string with node,the diffraction problem of light.In addition,through the method of separation of variables,some partial differential equations are transformed into differential operator problems with eigen-parameter dependent boundary conditions(BCs).For example,the vibration problem of the string on a sliding rod.Therefore,it is very important to study differential operators with internal discontinuity and eigen-parameter dependent BCs.For internal discontinuity,we describe them by attaching transmission conditions(TCs)at the discontinuous points.The research content of this thesis includes two parts:the first part studies the dependence of eigenvalues on parameters of three kinds of differential operators with TCs or eigen-parameter dependent BCs;the second part investigates the inverse spectral problem of a non-selfa djoint Dirac operator with TCs and eigen-parameter dependent BCs.In the first part of this thesis,we use operator theory to study the dependence of eigenvalues on parameters of three kinds of differential operators with different conditions.Firstly,we study a fourth-order differential operator,where the BCs containing eigen-parameter at both endpoints.Due to the BCs contain eigen-parameter,such problems cannot be considered in a general Hilbert space.By using the direct sum space theory in functional analysis,a direct sum Hilbert space related to the problem is defined by combining eigen-parameter in BCs.In this space framework,the problem is transformed into an operator problem.Then,the characteristic function is constructed by defining the basic solution.By using the continuity theorem of the isolated zero of the entire function,the continuous dependence of eigenvalues on the coefficient functions and weight function of the equation and coefficients of the BCs is proved.By constructing the generalized Lagrange equality between two problems,it is proved that the eigenvalues with respect to these parameters are differentiable in the sense of ordinary derivative or Frechet derivative,and the corresponding differential expressions are given.In particular,for the coefficients of the BCs,the differential expressions of the eigenvalues with respect to the coefficient matrices and each coefficient are given,respectively.To make the problem more general,we study the fourth-order discontinuous differential operator,where the eigen-parameter dependent BCs are more complex than those of the previous problem.Compared to the previous problem,due to the BCs are different and the TCs appear in the problem,we need to define a Hilbert space related to the discontinuous problem.In this space,an operator related to the discontinuous problem is defined.Further,the differential expressions of the eigenvalues with respect to the parameters are obtained.In particular,the coefficients of the TCs are also included in these parameters.Finally,we study a discontinuous Sturm-Liouville(S-L)problem with distribution potentials,where the BCs are divided into separated,real coupled,and complex coupled.In order to obtain the dependence of eigenvalues with respect to the parameters of the problem,it should be noted that due to the equations and conditions of this problem are different from the problems above,it is necessary to define a space suitable for the studied problem.In the second part of this thesis,the inverse spectral problem of a class of non-selfadjoint discontinuous Dirac operator is studied by method of spectral mappings,where there are finite discontinuous points and eigen-parameter dependent BCs and TCs.Firstly,the asymptotic estimates of the eigenvalues and the solutions of the equation are obtained through calculations.Secondly,by defining the generalized norming constants corresponding to the operator,the relationship between the generalized norming constants and other spectral characteristics is given,and then the uniqueness theorem of the inverse problem is proved using the Weyl function and generalized spectral data,respectively.Finally,two reconstruction algorithms are provided.This thesis is divided into five chapters:Chapter 1 summarizes the physical background,research significance,and current development situations of this study,and introduces the main research content and results of this thesis.In Chapters 2 and 3,the dependence of eigenvalues of continuous and discontinuous fourth-order differential operators is studied.The BCs for both types of problems contain eigen-parameter.In Chapter 4,we study the dependence of eigenvalues of discontinuous S-L problem with distribution potentials on parameters is studied.In Chapter 5,we study the inverse spectral problem of a nonselfadjoint Dirac operator with finite discontinuous points,where both BCs and TCs contain eigen-parameter.