Acoustic phase lenses in superfluid helium as models of composite spacetimes in general relativity: Classical and quantum features

In spirit of the well-known analogy between inviscid fluids and pseudo-Riemannian manifolds we study spherical singular hypersurfaces in the static superfluid. Such hypersurfaces turn to be the interfaces dividing the superfluid into the pairs of spherical domains, for instance, phases “ superfluid A superfluid B” or “impurity superfluid”. It is shown that these shells form the acoustic lenses which have to be sonic counterparts of the usual optical ones. The exact equations of motion of the lens interfaces are obtained. Also some quantum aspects of the theory are considered, thereby we calculate energy spectra for bound states of the acoustic lenses in dynamical equilibrium, taking into account the spatial topology of both the black hole and wormhole type. PACS number(s): 04.40.Nr, 11.27.+d, 43.35.+d, 67.57.De