Abstract
We consider the quantum problem of a particle in either a spherical box or a finite spherical well confined by a circular cone with an apex angle 2θ₀ emanating from the center of the sphere, with 0<θ₀<π. This non-central potential can be solved by an extension of techniques used in spherically-symmetric problems. The angular parts of the eigenstates depend on azimuthal angle ϕ and polar angle θ as P_λ^m(cosθ) e^imϕ where P_λ^m is the associated Legendre function of integer order m and (usually noninteger) degree λ. There is an infinite discrete set of values λ=λ_i^m (i=0,1,3,…) that depend on m and θ₀. Each λ_i^m has an infinite sequence of eigenenergies E_n(λ_i^m), with corresponding radial parts of eigenfunctions. In a spherical box the discrete energy spectrum is determined by the zeros of the spherical Bessel functions. For several θ₀ we demonstrate the validity of Weyl's continuous estimate N_W for the exact number of states N up to energy E, and evaluate the fluctuations of N around N_W. We examine the behavior of bound states in a well of finite depth U₀, and find the critical value U_c(θ₀) when all bound states disappear. The radial part of the zero energy eigenstate outside the well is 1/r^λ+1, which is not square-integrable for λ≤ 1/2. (0<λ≤ 1/2 can appear for θ₀>θ_c≈ 0.726π and has no parallel in spherically-symmetric potentials.) Bound states have spatial extent ξ which diverges as a (possibly λ-dependent) power law as U₀ approaches the value where the eigenenergy of that state vanishes.