2021   04   en   p.53-64 Shakir M. Nagiyev1, Shovqiyya A. Amirova1, Hasan P. Veliyev2,
Comment on “a quantum exactly solvable nonlinear oscillator with quasi-harmonic behavior” and “algebraic solutions of shape-invariant position-dependent effective mass systems” and others


Papers [3, 4] are devoted to the study of the quantum version of the nonlinear classical harmonic oscillator proposed in [1]. The authors of [3, 4] applied various quantization schemes for the classical Hamiltonian and expressed the wave functions of a quantum nonlinear harmonic oscillator in terms of Λ-dependent Hermite polynomials Hn (y,Λ) and λ ̃- modified Hermite polynomials Hn (ζ,λ ̃ ), respectively. We showed that these polynomials are not new, but in fact, for Λ<0 and λ ̃<0 , are Gegenbauer polynomials Cnν (x), and for Λ>0 and λ ̃>0, they are special cases of pseudo-Jacobi polynomials Pn (x;ν,N) corresponding to the value of the parameter ν=0. In addition, we have constructed a generating function for the polynomials Pn (x;0,N) and established their connection with the polynomials Cnν (x), and also constructed two exactly solvable potentials associated with the pseudo-Jacobi polynomials.

Keywords: nonlinear harmonic oscillator, wave functions, Gegenbauer and pseudo-Jacobi polynomials, generating function, limit relations.
PACS: 03.65.-w Quantum mechanics, 02.30. Hq Ordinary differential equations, 03.65.Ge Solutions of wawe equatins: bound states


Received: 15.11.2021


1. Institute of Physics, Azerbaijan National, Academy of Sciences, H. Javid Avenue, 131, AZ-1143, Baku, Azerbaijan,
e-mail: hakir.m.nagiyev@gmail.com, shovqiyya@mail.ru
2. Azerbaijan Technical University, H. Javid Avenue, 25, AZ-1073, Baku, Azerbaijan, e-mail: hasan.veliyev.1948@gmail.com

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