Abstract
The spectrum and the solutions of the one-dimensional Schrödinger equation with a quasiperiodic potential V are investigated numerically. V is taken as the sum of two incommensurate periodic δ-function series. Quasiperiodicity, in contrast to periodicity, makes possible the existence of localized modes. The typical spectra consist of bands corresponding to extended solutions and points corresponding to localized ones. Extended solutions (three-frequency quasiperiodic) are found from the initial-value problem. Localized solutions (exponential decay on both sides) cannot be found in this way owing to numerical instability. Instead, they are obtained from a boundary-value method. An alternative method to find the point spectrum for bounded potentials is proved to be also valid for δ-function potentials, as used here. Connections with the magnetohydrodynamic spectrum of toroidal plasmas are pointed out.
- Received 9 July 1991
DOI:https://doi.org/10.1103/PhysRevA.45.1116
©1992 American Physical Society