Quantum particle in a spherical well confined by a cone

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 2θ ₀ emanating from the center of the sphere, with 0 < θ ₀ < 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 ${P}_{\lambda }^{m}(\cos \theta ){{\rm{e}}}^{{im}\varphi }$ where ${P}_{\lambda }^{m}$ is the associated Legendre function of integer order m and (usually noninteger) degree λ. There is an infinite discrete set of values $\lambda ={\lambda }_{i}^{m}$ (i = 0, 1, 3, ...) that depend on m and θ ₀. Each ${\lambda }_{i}^{m}$ has an infinite sequence of eigenenergies ${E}_{n}({\lambda }_{i}^{m})$ , 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 θ ₀ we demonstrate the validity of Weyl's continuous estimate ${{ \mathcal N }}_{W}$ for the exact number of states ${ \mathcal N }$ up to energy E, and evaluate the fluctuations of ${ \mathcal N }$ around ${{ \mathcal N }}_{W}$ . We examine the behavior of bound states in a well of finite depth U ₀, and find the critical value U _c (θ ₀) 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 θ ₀ > 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 U ₀ approaches the value where the eigenenergy of that state vanishes.