Classification of oscillators in the Heisenberg-matrix representation

Abstract

For an arbitrary finite and not too singular (presumably, phenomenological) superposition of potentials r^delta 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.