Yuan You

The quantum characteristics of a class of complicated double ring-shaped non-central potential

Firstly we outline how to obtain the normalized polar angular wave functions of the Schrodinger equation with a class of complicated double ring-shaped non-central potential by introducing the super-universal associated Legendre polynomials, and present the exact energy equation and normalized radial wave functions for the given central potentials, such as the harmonic oscillator and Coulomb potential. We then discuss in detail the quantum characteristics of the polar and radial wave functions and energy equation, and their reducing problems. These features include bound state relations between the harmonic oscillator and Coulomb potential quantum systems, and degeneracy of their energy levels. We also observe that the bound state relations, including the energy levels and wave functions for the harmonic oscillator and Coulomb potential systems, can be obtained from each other by mapping the system parameters. Special cases related to reduction questions are discussed in detail.

Exact solutions to a class of differential equation and some new mathematical properties for the universal associated-Legendre polynomials

Abstract The exact solutions to a class of differential equation are studied. Some special cases are discussed for the central potentials, single ring-shaped potential and the angular Teukolsky equation. A new expression to the associated Legendre polynomials is found. Some new properties of the universal associated-Legendre polynomials (UALPs) including the generating function, Rodrigues’ formula, parity, some special values and the recurrence relations are presented. A new different Rodrigues’ formula of associated Legendre polynomials is also obtained.

Parity inversion property of the double ring-shaped oscillator in cylindrical coordinates

Using the functional analysis method, we present the exact solutions of the double ring-shaped oscillator (DRSO) potential with certain parity in the cylindrical coordinates. Such a quantum system is separated to two differential equations, i.e. a one-dimensional harmonic oscillator plus an inverse square term and a two-dimensional harmonic oscillator plus an inverse square term. The key point is how to find the adapted symmetrical solutions of the one-dimensional harmonic oscillator plus an inverse square term at the singular point z = 0. The obtained results are compared with those in the spherical coordinates. We also explore intimate connections ET(iρ) = −ET(ρ) and Ez(iz) = −Ez(z) by substituting ρ → iρ and z → iz.

Evaluation of more general integrals involving universal associated Legendre polynomials

We find that the solution of the polar angular differential equation can be written as the universal associated Legendre polynomials. We present a popular integral formula which includes universal associated Legendre polynomials and we also evaluate some important integrals involving the product of two universal associated Legendre polynomials Pl′m′(x), Pk′n′(x) and x2a(1−x2)−p−1, xb(1±x2)−p, and xc(1−x2)−p(1±x)−1, where l′≠k′ and m′≠n′. Their selection rules are also mentioned.