Abstract
We study an energy-dependent potential related to the Rosen–Morse potential. We give in closed-form the expression of a system of eigenfunctions of the Schrödinger operator in terms of a class of functions associated to a family of hypergeometric para-orthogonal polynomials on the unit circle. We also present modified relations of orthogonality and an asymptotic formula. Consequently, bound state solutions can be obtained for some values of the parameters that define the model. As a particular case, we obtain the symmetric trigonometric Rosen–Morse potential for which there exists an orthogonal basis of eigenstates in a Hilbert space. By comparing the existent solutions for the symmetric trigonometric Rosen–Morse potential, an identity involving Gegenbauer polynomials is obtained.
Highlights
An energy-dependent Schrödinger equation appears for the first time in relativistic quantum mechanics with the Pauli–Schrödinger equation, given by Pauli [1] in the description of the spectrum of an electron in the presence of a magnetic field
Further developments in relativistic and non relativistic quantum mechanics was made by many authors [2,3,4,5,6,7,8,9,10]
The description of systems of N bosons bound is considered in [12] and the Hamiltonian formulation of the relativistic many-body problem in [4,5,13] lead to energy dependent potential models
Summary
An energy-dependent Schrödinger equation appears for the first time in relativistic quantum mechanics with the Pauli–Schrödinger equation, given by Pauli [1] in the description of the spectrum of an electron in the presence of a magnetic field. In the present manuscript we study a quantum system with energy dependent potential related to the Rosen–Morse trigonometric potential, used in describing the interatomic interaction of linear molecules and for describing polyatomic vibration states and energies of the NH3 molecule [18]. As a particular case, the symmetric trigonometric Rosen–Morse potential In such case, the solutions reduce to an orthogonal basis of eigenfunctions defined in a Hilbert space and are expressed in terms of the Gegenbauer or ultraspherical polynomials. The solutions reduce to an orthogonal basis of eigenfunctions defined in a Hilbert space and are expressed in terms of the Gegenbauer or ultraspherical polynomials By comparing this solution with other solutions given in the literature we obtain, as a consequence, an identity involving Gegenbauer polynomials.
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