Classification of oscillators in the Heisenberg-matrix representation

For an arbitrary finite and not too singular (presumably, phenomenological) superposition of potentials rdelta with rational exponents we solve the old problem of conversion of the corresponding differential Schrodinger bound-state problem into its matrix equivalent with the minimalized number L of non-zero diagonals. The construction-with proofs-is performed via the standard power-series method in its non-Hermitean matrix (sometimes called Hill-determinant) version. All the simplest and (in this sense) 'algebraizable' interactions are then displayed up to L=5. An example of application in perturbation theory is included.