Inverse Spectrum Problem Of Sturm-Liouville Differential Operator For Symmetric Potential Function

The inverse spectrum problem of the Sturm-Liouville system is mainly con-cerned with the problem of confirming uniquely and constructing the system ac-cording to spectral information.This problem is not only an important subject in applied mathematics research,but also has a wide and direct application in physics,meteorology,etc.In the thesis,firstly,we analyse the multiple and symmetrie property of the potential functions,then the uniqueness theorems of potential functions on the en-tire interval and partial interval are considered respectively and the reconstruction algorithm of potential functions is given.The main works are given as follows:In the first chapter,the physical background and situation of Stnrm-Liouville inverse spectrum problem are summarized,and then we introduce the main work of this thesis.In the second chapter,the uniqueness problems of Sturm-Liouville differential operators with multiple symmetric potential functions are studied.Under the con-dition that the potential functions satisfy the multiple symmetry,we can prove the uniqueness of the potential functions on entire interval according to the properties of the eigenvalue asymptotic formula;Secondly,the uniqueness problems of poten-tial functions on partial interval are considered:If the potential functions satisfy the multiple symmetry and are partially known,the corresponding uniqueness con-clusions are proved by using Marchenko's uniqueness theorem.Thus,the inverse spectrum theorems of Simon's mixed spectrum data are extended.In the third chapter,the reconstruction problem of Sturm-Liouville differential operators of symmetric potential functions is studied.Using the Levin interpolation formula,the potential functions under Dirichlet and Robin boundary value condi-tions are reconstructed respectively.