Shape invariance, raising and lowering operators in hypergeometric type equations

Abstract

The Schrödinger equations which are exactly solvable in terms of associated special functions are directly related to some self-adjoint operators defined in the theory of hypergeometric type equations. The fundamental formulae occurring in a supersymmetric approach to these Hamiltonians are consequences of some formulae concerning the general theory of associated special functions. We use this connection in order to obtain a general theory of Schrödinger equations exactly solvable in terms of associated special functions, and to extend certain results known in the case of some particular potentials.

Shape-invariant hypergeometric type operators with application to quantum mechanics

“…and the functions corresponding to consecutive values of m are related through the raising/lowering operators [2,3] …”

“…Table 1 allows one to pass in each case from our parameters α, β to the parameters used in a different approach. In [2,3], we presented a systematic study of the Schrödinger equations which are exactly solvable in terms of associated special functions. In the present paper, based on the factorization method [4,5] and certain results of Jafarizadeh and Fakhri [6], we extend our unified formalism by adding other shape-invariant operators.…”

Dynamical algebras for Pöschl–Teller Hamiltonian hierarchies

“…So, in order to circumvent this problem, a second way is to identify the eigenfunctions, including the normalization constants, in terms of some known special functions. Then, the ladder operators can be found by means of certain recurrence relations of these functions [7,[18][19][20][21][22][23]. For example, the ''dynamical potential algebras" of Pöschl-Teller (PT) potentials have been found by connecting its eigenfunctions with Wigner rotation matrices [24].…”

Systems of orthogonal polynomials defined by hypergeometric type equations with application to quantum mechanics

“…In our previous paper [2], we presented a unified view on the systems of orthogonal polynomials defined by equation (1) in the case σ(s) ∈ {1, s, 1 − s 2 }. We analyzed the associated special functions, and the corresponding raising/lowering operators, and we presented some applications to quantum mechanics.…”

“…In the present paper, we extend our unified formalism to all the cases presented in Table 1. This extension is obtained through small changes in certain proofs presented in [1,2], and is mainly based on our remark that the number of orthogonal polynomials defined by (1) depends on the set of all γ ∈ R for which…”