Sturm-Liouville Semi-inverse Problem With Multiple Discontinuities And Its Generalization

The Sturm-Liouville problem originated from the treatment of heat conduction by Fourier,and its spectrum and inverse spectrum problem has far-reaching theoretical implications for the development of differential operators and thus has received much attention.In addition,the research could address a variety of practical questions.Driven by applications to string vibration problems,acoustics problems,electronics problems,etc.,the problem has developed into a field of applied mathematics.The Sturm-Liouville problem with internal discontinuities originated from the study of the medium splitting problem.The Sturm-Liouville problem with internal discontinuities originated from the study of the partitioning of the medium.In this paper,we first discuss the Sturm-Liouville inverse spectrum problem with multiple discontinuities,and develop an algorithm to implement the potential function reconstruction.In this paper,we first discuss the inverse spectral problem of Sturm-Liouville with multiple discontinuities,and develop a reconstruction algorithm for realizing the potential function.The inverse spectra of the Sturm-Liouville operator with matrix values are considered,and the uniqueness theorem and the semi-inverse theorem of the potential function are given.The main contents of this paper are organized as follows:The first chapter introduces the background,the current state of research and the main contents of the paper on the Sturm-Liouville operator with multiple internal discontinuities and the matrix-valued Sturm-Liouville operator;The second chapter studies the reconstruction of the Sturm-Liouville operator with a finite number of internal discontinuities,and gives the reconstruction formula of the operator by means of a set of potential functions of the spectrum and half of the interval;The third chapter studies the inverse of the matrix-valued Sturm-Liouville operator,using Marchenko's uniqueness theorem,the uniqueness theorem of the operator and the corresponding reconstruction algorithm.