Quantum-mechanical confinement with the robin condition

The energy spectrum of a nonrelativistic quantum particle in the confinement state in a closed spatial volume at general boundary confinement conditions (the Robin conditions) is investigated. It is shown that the properties of such a state are substantially more nontrivial compared with particle confinement using the potential barrier. It is also shown for a hydrogen-like atom arranged in a spherical cavity with radius R that if the surface layer with nonzero depth d plays the role of the boundary of the confinement region, all the energy levels of a discrete spectrum of the atom have a finite limit at R→0, while the R-dependence of the lower layer at physically substantial parameters of the surface layer contains a deep well-pronounced minimum, in which the binding energy is considerably higher than for the lower 1s level of a free atom.