The origin of a flat band

The origin of a 'flat band' as defined in the 'flat band-steep band' model for superconductivity has been studied. A 'local inversion center' is derived mathematically as the necessary and sufficient condition for an occurrence of a 'flat band' at an arbitrary k→ point. From the k→-dependent part of E(k→j), a 'pseudo-inversion center' is demonstrated to be a sufficient condition based on the Patterson symmetry. The chemical bonding origin of a 'flat band' is also studied analytically with a tight-binding approach