Inverse eigenvalue problem for discrete three-diagonal Schroedinger operator and the continuum limit in Tikhonov regularization approach

Abstract

In the present article the self-contained derivation of main formulas of the inverse eigenvalue problem for the Sturm-Liouville operator is given on the base of its discrete approximation and the subsequent transition to the limit of the continuum coordinate. In doing so, we adhere to Tikhonov concept, where the discrete approximation is performed as a regularization of the ill-posed operator problem. While the continuum limit for the three-diagonal matrix Hamiltonian is impossible as such, following the Tikhonov procedure leads not to a limit for the potential matrix itself, but rather for a certain combination of the matrix elements, and this limit is precisely the desired potential in the continuum case.