The Heun Equation and Generalized Sl(2) Algebra

Abstract

In this paper, first we introduce the Heun equation. In order to solve such equation we show the generators of generalized sl(2). Second, we arrange the Heun equation in terms of new operators formed of generalized sl(2) generators and it's commutator relation. Here, instead of J + (r), J − (r) and J 0 we use the P + (r), P − (r) and P 0 (r) as operators of generalized sl(2) algebra. This correspondence gives us opportunity to arrange the parameters α and β in P 0 (r). Also, the commutator of such operators leads us to have generalized sl(2) algebra. Also, we obtain the Casimir operators and show that it corresponds to P + , P − and some constants. These operators lead to deform the structure of generalized sl(2) algebra in the Heun equation. Finally, we investigate the condition for exactly and quasi-exactly solvable system with constraint on the corresponding operators P + and P − .

Polarization and stratification of axionically active plasma in a dyon magnetosphere

“…This means that in the model under consideration the pseudoscalar fieldΦ is described by the Fuchstype function with the parameters (81). Also this function can be indicated as generalized Heun's function (see, e.g., [49] and references therein); the corresponding equations are well studied in the context of searching for generalized spherical wave functions. Typical sketches for the fundamental solutions Y (1) (x) and Y (2) (x) are presented in Figs.1 and 2.…”

Magnetoelectrostatics of axionically active systems: Induced field restructuring in magnetic stars

“…In the presented work we made the first step in the analysis of the magnetic field restructuring driven by the interaction of the dipolar magnetic field with the axionic dark matter, which surrounds the star. We analyzed the solutions to the equations of magnetoelectro-axiono-statics in the static gravitational field of the Reissner-Nordström type; it was shown that the key equations for this model can be reduced to the Heun and Fuchs equations, which are known in mathematical physics [34][35][36][37]. Important detail of this analysis is that three dimensionless guiding parameters of the model happen to be encoded in the structure of the distorted halo of the axionic dark matter: we mean the parameter a = r− r+ , describing the relative depth of the inner and outer horizons of the magnetic star, and the parameters ω and Ω introduced by (80), which describe the effectiveness of the axion-photon coupling at the presence of the dipolar magnetic field.…”

“…Let us compare (66) with the Heun equation [34,35], which is a particular case of the known Fuchs equation [36,37]:…”

New application of the Killing vector field formalism: modified periodic potential and two-level profiles of the axionic dark matter distribution

“…(see, e.g., [51,52]), which is in its turn the particular case of the Fuchs equation [53,54]. The solution of this equation is regular at infinity, when ǫ+γ+δ = α+β+1.…”