The positioning of a bubble inside a many fermion system does not affect the volume, surface or curvature terms in the liquid drop expansion of the total energy. Besides possible Coulomb effects, the only other contribution to the ground state energy of such a system arises from quantum effects, often called in nuclear physics shell correction energy effects. Related phenomena could be observed in infinite systems. Homogeneous neutron matter at subnuclear densities becomes unstable towards the formation of inhomogeneities. Depending on the average value of the neutron one can observe the appearance of either bubbles, rods, tubes or plates embedded in a neutron gas. We estimate the quantum corrections to the ground state energy (which could be termed either shell correction or Casimir energy) of such phases of neutron matter. The calculations are performed by evaluating the contribution of the shortest periodic orbits in the Gutzwiller trace formula for the density of states. The magnitude of the quantum corrections to the ground state energy of neutron matter are of the same order as the energy differences between various phases and thus can lead to changes in the sequence of the nuclear shape transitions in the neutron star crust. We discuss also the dependence of these corrections on a number of physical parameters (density, filling factor, temperature, lattice distortions). Similar phenomena are expected to occur as well in atomic clusters, condensed matter systems, dilute atomic fermi condensates and quark--gluon plasma.