Quantum confinement under Neumann condition: atomic H filled in a lattice of cavities

Energy spectrums of a nonrelativistic particle and an H-like atom in a spherical box of size R with general conditions of "not going out" through the box surface are explored. The lowest energy levels reconstruction is described from the point of view of their asymptotical behavior for large R. The role of von Neumann-Wigner level reflection/avoided crossing effect in this spectrum reconstruction is emphasized. The properties of atomic H ground state in a cell, formed by a spherical cavity with an outer potential shell and Neumann condition on the outward boundary, are studied in detail. Some of them turn out to be quite new. The relevance of such a cell to a cubic lattice of cavities, occupied by H, is discussed be means of first principles and assumptions of the Wigner-Seitz model.