We consider the problem of a quantum particle in a spherical potential well confined by a cone. By varying the apex angle 2θ₀ of the cone we modify the confinement of the particle from being inside a needle-like space via hemisphere and to an almost complete sphere with an excluded needle-like volume.  In an infinite spherical well the energy spectrum is discrete and is determined by the zeros of the spherical Bessel functions, whose order is determined by θ₀. We numerically demonstrated, for several values of θ₀, that the total number of eigenstates up to energy E is correctly given by Weyl’s formula [1].

For the spherical well of a finite depth -U, there is a discrete energy spectrum of bound states with eigenenergies E<0 and a continuous spectrum of states with E>0, as long as the well is deep enough. For each θ₀ there is a critical depth U_c at which the last bound state disappears. The bound states are localized inside the well such that outside the well the wavefunctions decay over a length scale (localization length) ξ. Close to the critical depth U_c the localization length of the last remaining bound state diverges as ξ~(U-U_c)^-ν, with exponent ν=1/2 for small and moderate cone apex semi-angles.  For θ₀ >0.726π, the exponent ν depends on θ_0, reaching 1 for θ₀ = π.

[1] V. Ivrii, Bull. Math. Sci. 6 (2016).