New theory of periodical systems by finite interval inverse problem

We show that the mechanism of gap formation has a resonance nature. The special real fundamental solutions were discovered which `paradoxically' have knot distribution with a period coinciding with that of potential at all energies of the whole lacuna interval. In terms of these solutions resonance gap appearance gets the most direct explanation: ever repeating hits by the potential result in exponential increase (decrease) of the wave amplitudes in the forbidden zones. The analogous alternating hits from opposite sides are responsible for the wave beatings in allowed zones. The inversion technique gives rise to zone control algorithms - shifting chosen boundaries of spectral bands, changing degree of zone forbiddenness. All this cannot be achieved by the previous Bloch theory.