Electronic structure of a quasiperiodic system

The general solution of a Schrödinger equation with a quasiperiodic potential in $n$ dimensions is obtained. A boost technique is presented, which will transform the problem to the solution of a periodic pseudo-Schrödinger equation in $n+m$ dimensions, to which Floquet-Bloch theory is applicable. We show that the eigenfunctions of the original problem and the boosted problem are related to each other by a simple radon transform, and the eigenvalues are exactly equal. We identify the hierarchical gap structure in the energy spectrum observed in numerical simulations and we show that the location of gaps can be indexed by the reciprocal wave vector given by the diffraction pattern of the quasicrystal. The position and the magnitude of the gaps so predicted are in qualitative agreement with numerical simulations.