Dong-Sheng Sun

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.

Approximate analytical solutions of scattering states for Klein-Gordon equation with Hulthén potentials for nonzero angular momentum

In this paper, using the exponential function transformation approach along with an approximation for the centrifugal potential, the radial Klein-Gordon equation with the vector and scalar Hulthén potential is transformed to a hypergeometric differential equation. The approximate analytical solutions of t-waves scattering states are presented. The normalized wave functions expressed in terms of hypergeometric functions of scattering states on the “k/2π scale” and the calculation formula of phase shifts are given. The physical meaning of the approximate analytical solution is discussed.