Spectrum and bandwidth of an exactly soluble incommensurate model

Abstract

The structure of the energy spectrum of the exactly soluble Maryland model is examined in detail for both incommensurate and commensurate modulations. The study is motivated by a formal similarity with Harper's equation describing crystal electrons in a uniform magnetic field. An exact expression is derived for the characteristic polynomial which determines the energy eigenvalues, allowing explicit verification of the transition from band structure, for commensurate modulations, to a point spectrum in the commensurate limit. An application is the calculation of the total width of the bands. A semiclassical argument is used to assess the relevance of the results to models of physical interest.