Energy bands and Wannier functions of the fractional Kronig-Penney model

Energy bands and Wannier functions of the fractional Schrödinger equation with a periodic potential are calculated. The kinetic energy contains a Riesz derivative of order α, with 1 < α ≤ 2, and numerical results are obtained for the Kronig-Penney model. Bloch and Wannier functions show cusps in real space that become sharper as α decreases. Energy bands and Bloch functions are smooth in reciprocal space, except at the Γ point. Depending on symmetry, each Wannier function decays as a power-law with exponent −(α+1) or −(α+2). Closed forms of their asymptotic behaviors are given. Each higher band displays anomalous behavior as a function of potential strength. It first narrows, becoming almost flat, then widens, with its width tending to a constant. The position uncertainty of each Wannier function follows a similar trend.