Inverse Sturm-liouville Problems Of Atkinson Type

The inverse spectral problems of Sturm-Liouville(S-L)problems were first proposed by V.A.Ambarzumian in 1930 s.The inverse spectral theory of classical S-L problems have wide application domain in mechanics and vibration model,physics,quantum mechanics and other fields,so this theory has attracted lots of researchers' attention.In recent years,a series of essential research results have been obtained on the inverse spectral theory of classical S-L problems,which make the inverse spectral problems of S-L problems developed rapidly.However,some scholars are not limited in the study of the inverse spectral theory of classical S-L problems,they studied the inverse spectral theory of S-L problems of Atkinson type.Nowadays,the S-L problems of Atkinson type have been made some theoretical achievements.However,the inverse spectral problems are not perfect yet.For example,the inverse spectral problems of S-L problems with transmission conditions,the inverse spectral problems of S-L problems with eigenparameter-dependent boundary conditions,the inverse spectral problems of higher order boundary value problems and the S-L problems with distribution potential function,etc.The problems mention above have not conclusions yet.In view of the above situation,this paper mainly focuses on a class of special and essential problems(the inverse spectral S-L problems of Atkinson type)in the spectral theory of differential operators,using the matrix representations of the boundary value problems and the corresponding inverse matrix eigenvalue problems,we obtain the conclusions of the inverse spectral problems of some boundary value problems.Firstly,the second-order S-L problems with transmission conditions are discussed.Due to the existence of the transmission condition,the matrix representation is a generalized Jacobi matrix or a generalized cyclic Jacobi matrix with the same sign of subdiagonal but not completely symmetric.We determine the number of subdiagonal items with the same sign but with incomplete symmetry,give the conditions of transmission matrix,and obtain the conclusions of inverse spectral problems.Secondly,the second-order S-L problems with eigenparameter-dependent boundary conditions arediscussed.Due to the existence of the eigenparameter-dependent boundary conditions,the matrix representation is a generalized pseudo-Jacobi matrix or a generalized cyclic pseudo-Jacobi matrix with the not exactly same sign of subdiagonal.We determine the spectral parameters of the boundary conditions,and obtain the conclusions of inverse spectral problems.Then,some kinds of the fourth-order boundary value problems of Atkinson type are discussed.We can use the conclusions of the inverse eigenvalue problem of banded matrices,then reconstruct the block Jacobi matrices,and obtain the conclusions of the inverse spectral problems.Finally,the second-order S-L problems with distribution potential function are discussed.We use the conclusions of inverse eigenvalue problems of Jacobi matrices and cyclic Jacobi matrices,then reconstruct the matrix representations of Jacobi matrix and cyclic Jacobi matrix,and obtain the conclusions of the inverse spectral problems.