The Inverse Sturm-Liouville Problem By Three Spectra And With The Parameter In The Boundary Conditions

The inverse spectral problem of Sturm-Liouville systems is mainly concerned with the uniqueness and reconstruction of the systems under the condition that mixed spectral data is given.The researches of this problem not only play an important role in mathematics but also have a wide and direct application in physics and natural science.Therefore,it has aroused great interest and high attention to mathematicians and physicists,which made this problem become one of the popular researches in applied mathematics.In this thesis,the inverse Sturm-Liouville problem by three spectra is stud-ied first then for the inverse Sturm-Liouville problem with the parameter in the boundary conditions,the uniqueness theorems are correspondingly proved and the reconstruction algorithms are given by the corresponding spectra data.The main works are given as follows:In the first chapter,firstly we sum the research backgrounds,significance and advance of the inverse Sturm-Liouville problem by three spectra and with the pa-rameter in the boundary conditions,then introduce the main work of this thesis.In the second chapter,the inverse Sturm-Liouville problem by three spectra is considered.We prove that if the given two sequences can divided into three sequences in certain conditions,which can be the corresponding parts of eigenvalues of three Sturm-Liouville problems,then the potential function on the entire interval can be uniquely determined by the corresponding parts of the three spectra.Further,we establish the algorithm for Gelfand-Levitan integral equation to the unknown potential.In the third chapter,the inverse Sturm-Liouville problem with the parameter in the boundary conditions is considered.Making use of the Residue theorem,we prove that one spectrum of right boundary dependence on the parameter A3 can uniquely determine the potential on entire interval,and we establish the algorithm for Marchenko integral equation to the unknown potential.Moreover,we consider two inverse Sturm-Liouville problems with the parameter in the both left and right conditions,the uniqueness theorems are correspondingly given.