One-Dimensional Schrödinger Equation with an Almost Periodic Potential

Stellan Ostlund; Rahul Pandit; David Rand; Hans Joachim Schellnhuber; Eric D. Siggia

doi:10.1103/PhysRevLett.50.1873

One-Dimensional Schrödinger Equation with an Almost Periodic Potential

Stellan Ostlund, Rahul Pandit, David Rand, Hans Joachim Schellnhuber, and Eric D. Siggia

Phys. Rev. Lett. 50, 1873 – Published 6 June 1983

Abstract

Recent theories of scaling in quasiperiodic dynamical systems are applied to the behavior of a particle in an almost periodic potential. A special tight-binding model is solved exactly by a renormalization group whose fixed points determine the scaling properties of both the energy spectrum and certain features of the eigenstates. Similar results are found empirically for Harper's equation. In addition to ordinary extended and localized states, "critical" states are found which are neither extended nor localized according to conventional criteria.

Received 8 February 1983

DOI:https://doi.org/10.1103/PhysRevLett.50.1873

Authors & Affiliations

Stellan Ostlund^*

Institute for Theoretical Physics, University of California, Santa Barbara, California 93106

Rahul Pandit

Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853

David Rand

Mathematics Institute, University of Warwick, Coventry CV4 7AL, England

Hans Joachim Schellnhuber^†

Institute for Theoretical Physics, University of California, Santa Barbara, California 93106

Eric D. Siggia

Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853

^*Address after 1 September 1983; Physics Department, University of Pennsylvania, Philadelphia, Pa. 19104.
^†Present address: Fachbereich Physik, Universität Oldenburg, Oldenburg, West Germany.