Abstract
With the help of methods developed for the Jacobi continued fraction and the Stieltjes continued fraction, we investigate some inverse spectral problems related to the Jacobi matrix equation and the Stieltjes string equation, which may be viewed respectively as the discrete analogues of the potential equation and the string equation studied in the classical theory of Sturm-Liouville equations. We prove, among others, a Dirichlet-Neumann-isospectral theorem for Stieltjes string equations, we find a necessary and sufficient condition for the transformability of a Jacobi matrix equation into a Stieltjes string equation and provide a transformation method. We investigate a theory of Jacobi matricial couples which is related to the two spectra Borg's theorem in the theory of Sturm-Liouville equations. We also consider some inverse spectral problems related to persymmetric Jacobi matrices and even Stieltjes strings with prescribed eigenvalues.
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