  We study singular Schrodinger operators with an attractive interaction
supported by a closed smooth surface A in R^3 and analyze their behavior in the
vicinity of the critical situation where such an operator has empty discrete
spectrum and a threshold resonance. In particular, we show that if A is a
sphere and the critical coupling is constant over it, any sufficiently small
smooth area preserving radial deformation gives rise to isolated eigenvalues.
On the other hand, the discrete spectrum may be empty for general deformations.
We also derive a related inequality for capacities associated with such
surfaces.
