Metallic Nanosphere in a Magnetic Field: an Exact Solution

We consider the electron gas moving on the surface of a sphere in a uniform magnetic field. An exact solution of the problem is found in terms of oblate spheroidal functions, depending on the parameter $p= \Phi/\Phi_0$, the number of flux quanta piercing the sphere. The regimes of weak and strong fields are discussed, the Green's functions are found for both limiting cases in the closed form. In the weak fields the magnetic susceptibility reveals a set of jumps at half-integer $p$. The strong field regime is characterized by the formation of Landau levels and localization of the electron states near the poles of the sphere defined by a direction of the field. The effects of coherence within the sphere are lost when its radius exceeds the mean-free path.