Inverse eigenvalue problem for the discrete three-diagonal Sturm-Liouville operator and the continuum limit

摘要

The self-contained derivation of the inverse eigenvalue problem is given using a discrete approximation of the Sturm-Liouville operator on a bounded interval. Within this approximation, the Hamiltonian is treated as a finite three-diagonal symmetric Jacobi matrix. This derivation is more correct in comparison with previous works which used only single-diagonal matrix. It is demonstrated that the inverse problem procedure is nothing but the well-known Gram-Schmidt orthonormalization in Euclidean space for special vectors numbered by the space coordinate index. All the results of the usual inverse problem with continuous coordinate are reobtained by employing a limiting procedure, including the Goursat problem-the equation in partial derivatives for the solutions of the inversion integral equation.