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

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  • Vellasco-Gomes, Arianne
  • de Figueiredo Camargo, Rubens
  • Bruno-Alfonso, Alexys

Abstract

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.

Suggested Citation

  • Vellasco-Gomes, Arianne & de Figueiredo Camargo, Rubens & Bruno-Alfonso, Alexys, 2020. "Energy bands and Wannier functions of the fractional Kronig-Penney model," Applied Mathematics and Computation, Elsevier, vol. 380(C).
    • Handle: RePEc:eee:apmaco:v:380:y:2020:i:c:s0096300320302356
    DOI: 10.1016/j.amc.2020.125266
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    1. Fernandez, Arran & Özarslan, Mehmet Ali & Baleanu, Dumitru, 2019. "On fractional calculus with general analytic kernels," Applied Mathematics and Computation, Elsevier, vol. 354(C), pages 248-265.
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