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 $2\theta_0$ emanating from the center of the sphere, with $0<\theta_0<\pi$. 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 $\varphi$ and polar angle $\theta$ 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 $\lambda$. There is an infinite discrete set of values $\lambda=\lambda_i^m$ ($i=0,1,3,\dots$) that depend on $m$ and $\theta_0$. 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 $\theta_0$ 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 $U_c(\theta_0)$ when all bound states disappear. The radial part of the zero energy eigenstate outside the well is $1/r^{\lambda+1}$, which is not square-integrable for $\lambda\le 1/2$. ($0<\lambda\le 1/2$ can appear for $\theta_0>\theta_c\approx 0.726\pi$ and has no parallel in spherically-symmetric potentials.) Bound states have spatial extent $\xi$ which diverges as a (possibly $\lambda$-dependent) power law as $U_0$ approaches the value where the eigenenergy of that state vanishes.