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 Pλm(cosθ)eimφ where Pλm is the associated Legendre function of integer order m and (usually noninteger) degree λ. There is an infinite discrete set of values λ=λim (i = 0, 1, 3, ...) that depend on m and θ 0. Each λim has an infinite sequence of eigenenergies En(λim), 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 NW for the exact number of states N up to energy E, and evaluate the fluctuations of N around NW . 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 ζ which diverges as a (possibly λ-dependent) power law as U 0 approaches the value where the eigenenergy of that state vanishes.