Here the authors by using the factorization method, construct finite- difference Schrödinger operators (Jacobi matrices) whose discrete spectra are composed from independent arithmetic, or geometric series. These systems originate from the periodic, or \(q\)-periodic closure of a chain of corresponding Darboux transformations. The Charlier, Krawtchouk, Meixner orthogonal polynomials, their \(q\)-analogs, and some other classical polynomials appear as the simplest examples for \(N=1,2\) where \(N\) is the period of closure.