Generalized anti-centrifugal potential

We generalize the quantum anti-centrifugal potential in the two-dimensional Euclidean plane to two-dimensional surfaces embedded in three-dimensional Euclidean space. We consider the sphere with two caps removed in some detail. We show that quantum particles in this space are “pushed” towards either of the cap boundaries. We also consider the two-dimensional Euclidean plane with an elliptic area removed and compute the quantum anti-centrifugal potential on the elliptic boundary. It is argued that a sufficiently thin electrically conducting nano-wire shaped as an ellipse will exhibit an inhomogeneous charge distribution due to this quantum potential.