Construction of dynamical symmetry group of the relativistic harmonic oscillator by the Infeld-Hull factorization method

Factorization method for difference equations of hypergeometric type on nonuniform lattices

“…The classical FM was based on the existence of a so-called raising and lowering operators for the corresponding equation that allows to find the explicit solutions in a very easy way. Going further, Atakishiyev and coauthors [5,9,6] have found the dynamical symmetry algebra related with the FM and the differential or difference equations. Of special interest was the paper by Smirnov [15] in which the equivalence of the FM and the Nikiforov et all theory [25] was shown, furthermore this paper pointed out that the aforementioned equivalence remains valid also for the nonuniform lattices that was shown later on in [20,29].…”

“…For more recent works see e.g. [5,6,11,29,30] and references therein. The classical FM was based on the existence of a so-called raising and lowering operators for the corresponding equation that allows to find the explicit solutions in a very easy way.…”

Recurrence relations for radial wavefunctions for theNth-dimensional oscillators and hydrogenlike atoms

“…Next we will establish a linear relation involving two radial functions of the IHO and the derivative of one of them. Some of these relations will define the so-called ladder operators for the radial wavefunctions and have important applications in the so-called factorization method (see, e.g., [3,12,13,18,24]). Theorem 3.2.…”

“…There are many applications in modern physics that require knowledge of the wavefunctions of hydrogenlike atoms and isotropic harmonic oscillators,especially for finding the corresponding matrix elements (see, e.g., [18,23] and references therein). There are several methods for generating such wavefunctions among which the so-called factorization method of Infeld and Hull [12] is of particular significance (for more recent papers see, e.g., [3,13,17,24]). Moreover, the recurrence relations and the ladder-type operators for these wavefunctions are useful for finding the transition probabilities and evaluation of certain integrals [18,23].…”

Factorization of the hypergeometric-type difference equation on non-uniform lattices: dynamical algebra

“…In this paper we wish to make one step further by studying the dynamical symmetry algebra for the hypergeometric-type difference equation ( 1) on the non-uniform lattices (2). Our approach is essentially based on the simple observation, formulated in [8]: In order to factorize an arbitrary difference equation, one should express it explicitly in terms of the shift (or displacement) operators exp(a d ds ), which are defined as exp(a d ds ) f (s) = f (s + a), a is some constant. For example, in the case of the equation (1) this corresponds to the substitutions ∆ = exp( d ds ) − 1 and ∇ = 1 − exp(− d ds ).…”