Polynomial Heisenberg algebras and higher order supersymmetry

It is shown that the higher order supersymmetric partners of the harmonic oscillator Hamiltonian provide the simplest non-trivial realizations of the polynomial Heisenberg algebras. A linearized version of the corresponding annihilation and creation operator leads to a Fock representation which is the same as for the harmonic oscillator Hamiltonian. 1. Introduction. The non-linear algebras have attracted attention in recent years due to the fact that they start to be applied in physics. Those structures arise as deformations of a Lie algebra in which some commutation relations are replaced by non-linear functions of the generators [1-5]. Of particular interest are the simplest deformations of the standard Heisenberg-Weyl algebra (the so-called polynomial Heisenberg algebras) in which the commutator of the annihilation and creation operators becomes a polynomial in the Hamiltonian but at the same time it commutes with those ladder operators as in the standard case [6-11]. Let us notice that these modified algebraic relations provide information about the spectrum, which turns out to be a variant of the equally spaced levels of the harmonic oscillator. Concrete realizations of those algebras arise if the annihilation and creation operators become differential operators of order greater than one [6,12]. The simplest non-trivial case of such a realization is provided by considering the natural pair of annihilation and creation operators for the Hamiltonians generated by means of the higher order supersymmetric quantum mechanics (HSUSY QM) applied to the harmonic oscillator [11]. This will be the main subject discussed in this paper. In order to do that, firstly we will present some generalities of the polynomial Heisenberg algebras. Then, we will proceed by discussing the standard SUSY and HSUSY QM as a mechanism to generate solvable partner potentials from a given initial one [11,13-26]. From that procedure it will be obvious the existence and the corresponding structure of a pair of natural creation and annihilation operators for the HSUSY partners of the oscillator potential, a very original construction proposed by Mielnik for the first order SUSY [27-28]. It will be shown that those operators realize in a simple way the polynomial Heisenberg algebras. Later on it will be constructed a 'linearized' version of those algebras in which the modified ladder operators act onto the energy eigenstates as the standard generators do in the Heisenberg-Weyl linear case [11,29]. We will finish the paper with some general conclusions and an outlook for future work.