Summary: We study singular Schrödinger operators with an attractive interaction supported by a closed smooth surface \(A\subset \mathbb R^3\) 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. {
©2009 American Institute of Physics}