Inverse Problems Of A Class Of Mixed Oscillating Systems

The study of mixed oscillating systems has very important effect in other disciplines and engineering and technical fields.In recent decades,driven by its applications,the study of the inverse problems for mixed oscillating systems has got broad attention from many scholars.As an important subject of inverse problems,inverse spectral problem not only has direct and extensive applications in the quantum physics,the eleetrology,the meteorology and other fields,but also is one of the valid ways for solving the solutions of non-linear equations in mathematical physics.In this paper,we systematically study the inverse spectral problems of a class of mixed oscillating systems,that is one-dimensional Schr?dinger operator with transmission conditions.We give the uniqueness theorems from two spectra,three spectra and Weyl function,and provide algorithms for the solutions of the inverse problems.We also consider inverse spectral problems of the Dirac-type operator with transmission conditions.The main works are given as follows:In the first chapter,we give a summary of the physical backgrounds of mixed oscillating systems;taking one-dimensional Schr?dinger operator as the main object,and elaborate the research advances of the inverse problems.In the second chapter,direct and inverse spectral problems for the one-dimensional Schr?dinger operator with transmission conditions are considered.We construct an operator whose spectrum is consistent with the operator,and prove the self-adjointness.Then we give the asymptotics of the solutions and eigenvalues.Further,we provide several uniqueness results for this inverse spectral problem from Weyl function,from spectral data and from two spectra.In the third chapter,three spectra inverse problems for the above mentioned operator are considered.We consider the situation that three spectra have overlapping eigenvalues,and prove that the uniqueness theorem remains valid by providing the normalizing constants corresponding to the overlapping eigenvalues.At the same time,we realize the reconstruction algorithms of the potential.In the fourth chapter,the uniqueness problems of the above mentioned operator are considered with parts of potential and parameters of the transmission conditions known.We prove that given half of information of potential and the parameters of transmission conditions,one spectrum can determine the operator,when the given parameters are more than half,finite eigenvalues can be missing.In the fifth chapter,the local Borg-Marchenko uniqueness theorems are considered.We establish the asymptotics of the difference of two Weyl functions corresponding to two Schr?dinger operators with the local information of potential and parameters of transmission conditions equal.We also consider the local uniqueness theorem for one-dimensional Schr?dinger operator with potential locally smooth.In the sixth chapter,we generalize the above results to the Dirac-type operator with transmission conditions and obtain the corresponding uniqueness theorems.