On an inverse problem for two spectra of finite Jacobi matrices (Q433308)

The author solves a version of the inverse spectral problem for two spectra of finite order real Jacobi matrices: the reconstruction of the matrix using two sets of eigenvalues, one for the original Jacobi matrix and one for the matrix obtained by replacing the last diagonal element of the Jacobi matrix by some other number.NEWLINENEWLINEThe first section is an introductory one.NEWLINENEWLINEThe second section presents the solution of the inverse problem for finite Jacobi matrices in terms of the eigenvalues and normalizing numbers. Thus the author concludes that the inverse problem with respect to the spectral data is solved uniquely up to the signs of the off-diagonal elements of the recovered Jacobi matrix.NEWLINENEWLINEIn the third section the author solves the main formulated problem. He reduces the inverse problem for two spectra to the inverse problem for eigenvalues and normalizing numbers solved in the second section and uses some formulae which express the normalizing numbers of a finite Jacobi matrix in terms of two its spectra. An explicit procedure of reconstruction of the matrix from the two spectra is given.