Abstract: We formulate a momentum space representation of an incommensurate linear Schrödinger equation. We use the representation to find algorithms for a number of fundamental electronic observables such as the local density of states of spatial configurations, total density of states, and the local density of states in momentum space, which is a parallel object to electronic band structure in the absence of periodicity. We further prove the equivalence of the density of states in the plane-wave formulation to that of the density of states of the real space Schrödinger equation through a properly averaged thermodynamic limit.

The methodology relies on tracking the plane-wave scattering between incommensurate potentials and using these ‘hopping’ parameters to construct a matrix describing the coupling of all interacting plane-waves, which we find to be indexed by a four-dimensional lattice. The algorithm relies on truncation of the matrix via an energy truncation and hopping distance truncation with rigorous convergence rates.

About the speaker

Daniel Massatt is an Assistant Professor in the Department of Mathematical Sciences at the New Jersey Institute of Technology in Newark, New Jersey. His research interests include Numerical Analysis, Quantum Theory, Multiscale Modeling, Spectral Theory, Topological Insulators, Incommensurate Materials, and Electronic Structure. Prior to his current position, he was an Assistant Professor in the Department of Mathematics at Louisiana State University in Baton Rouge, Louisiana.