Bose-Einstein condensation on axially symmetric surfaces

We investigate the phenomenon of Bose-Einstein condensation in ideal bosonic gases confined to axially symmetric surfaces of revolution. The single-particle Schr"odinger equation is formulated on a general surface and then explicitly solved in the ellipsoidal and toroidal geometries to determine the one-body energy spectrum. We discuss how the curved geometry impacts the quantum statistical properties of ideal Bose gases confined on these surfaces. Specifically, we observe that Bose-Einstein condensation is suppressed when the surface aspect ratio is increased and, correspondingly, it becomes highly elongated and acquires a one-dimensional character. We also evaluate the Bogoliubov excitation spectrum, providing insights into the collective excitations of the condensate. Our results establish the conditions to achieve quantum degeneracy in curved manifolds, thus guiding forthcoming experiments with thin shells, and set the basis for solving the few-to-many body problem in general surfaces of revolution.