标题： d维空间中受限和非受限 -a/r + b r² 势的离散谱 显示英文标题

标题： Discrete spectra for confined and unconfined -a/r + b r^2 potentials in d dimensions

摘要： 求解d维薛定谔方程，d\geq 2，对于库仑势加谐振子势V(r)=-a/r+br^2，b>0且a\ne 0的精确解。势V(r)既在全部空间中考虑，又在不可穿透球盒半径R的球形约束条件下进行考虑。借助渐近迭代方法，在某些约束条件下获得精确解析解以及一般近似解。这些解表现出本征能量关于a、b和R的参数依赖性。波函数具有幂函数、指数函数和多项式的乘积形式。 为了达到我们的结果，求解二阶微分方程的多项式解的问题 (\sum _{i=0}^{k+2}a_{k+2,i}r^{k+2-i})y"+(\sum _{i=0}^{k+1}a_{k+1,i}r^{k+1-i})y'-(\sum _{i=0}^{k}\tau _{k,i}r^{k-i})y=0 对 k=0,1,2 是解决的。 显示英文摘要

摘要： Exact solutions to the d-dimensional Schroedinger equation, d\geq 2, for Coulomb plus harmonic oscillator potentials V(r)=-a/r+br^2, b>0 and a\ne 0 are obtained. The potential V(r) is considered both in all space, and under the condition of spherical confinement inside an impenetrable spherical box of radius R. With the aid of the asymptotic iteration method, the exact analytic solutions under certain constraints, and general approximate solutions, are obtained. These exhibit the parametric dependence of the eigenenergies on a, b, and R. The wave functions have the simple form of a product of a power function, an exponential function, and a polynomial. In order to achieve our results the question of determining the polynomial solutions of the second-order differential equation (\sum_{i=0}^{k+2}a_{k+2,i}r^{k+2-i})y"+(\sum_{i=0}^{k+1}a_{k+1,i}r^{k+1-i})y'-(\sum_{i=0}^{k}\tau_{k,i}r^{k-i})y=0 for k=0,1,2 is solved.