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 $P_λ^m(\cosθ){\rm e}^{imφ}$ 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 $λ=λ_i^m$ ($i=0,1,3,\dots$) that depend on $m$ and $θ_0$. 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 $θ_0$ we demonstrate the validity of Weyl's continuous estimate ${\cal N}_W$ for the exact number of states $\cal N$ up to energy $E$, and evaluate the fluctuations of $\cal N$ around ${\cal N}_W$. 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 $λ\le 1/2$. ($0<λ\le 1/2$ can appear for $θ_0>θ_c\approx 0.726π$ 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.

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