Determine algorithms for restoring spectral orthogonality

Develop algorithms for restoring the orthogonality relation among transformed regular solutions after the spectral parameters are changed in the discrete inverse eigenvalue problem for a finite tridiagonal Schrödinger Hamiltonian.

Background

The paper formulates the discrete direct problem as a mapping from the tridiagonal potential matrix to its spectral data, consisting of the eigenvalues and norming constants. Changing the spectral parameters generally violates the associated orthogonality relation, so the inverse procedure must restore orthogonality before reconstructing the transformed regular solutions and potential.

Although the paper uses standard Gram–Schmidt reorthogonalization for its derivation, it explicitly states that the corresponding algorithms for the required restoration were not yet known. This leaves open the development of suitable algorithms beyond the standard linear-algebraic reorthogonalization procedure.