Fractional calculus has become an essential framework in geophysics, optics, and biological systems to capture long-range correlations and anomalous transport. In this article, we extend the success of fractional calculus in physical models to explore a particle in a periodic potential, where the Schrödinger equation is extended to its fractional form. This framework enables us to study how the Lévy index $q$ governs the formation and inversion of energy bands, offering a pathway to engineer new physical behaviors and device functionalities by tuning $q$ in periodic quantum systems. We solve the fractional Schrödinger equation for periodic rectangular potentials of varying height $V_0$, barrier thickness $L$, and well width $W$ using an imaginary-time evolution algorithm, and supplement the discrete energy dispersion through Gaussian process regression. This analysis reveals a qualitative shift in the system's band structure at $q=2$, separating into distinct regimes of dispersion that define behavior for $q>2$ and $q<2$. For $q>2$, the energy bands undergo an inverting transformation as symmetric minima emerge within the first Brillouin zone and shift from $k=0$ toward $k=\pm \pi /a$ with increasing $q$. These degenerate minima define a Bloch-momentum qubit, suggesting an analog to valley degrees of freedom used in valleytronics. The $q$ at which the inversion completes scales as $q\propto V_0^{-0.28\pm 0.05}, q\propto L^{-0.35\pm 0.08}$, and $q\propto W^{-0.49\pm 0.06}$ when varying potential parameters individually, indicating a tunable transformation sensitivity to potential geometry. In contrast, for $q<2$, the ground band hardens around $k=0$, with a dispersion following the functional form of ${C\left|k\right|}^q+E_0$ near $k=0$. This suggests an effective mass of 0 for $1<q<2$ at the band's lowest energy state. These results demonstrate that the Lévy index serves as a tunable degree of freedom in quantum periodic systems, capable of driving band inversion, modulating the band gap, and reshaping carrier dynamics through effective-mass control.

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• Bifurcations
• Levy flights
• Quantum engineering