Novel behavior of the one-dimensional incommensurate Schrodinger equation when the potential is derived from a Kolmogorov-Arnol'd-Moser torus is found by exploitation of univeral scaling properties. Our numerical investigation reveals in particular that nearly torus-breaking electronic states undergo series of back-and-forth localization transitions in sensitive dependence on the potential strength.PACS numbers: 71.55.Jv, 71.30. + h Incommensurate systems may be conceived as being generated from higher-dimensional commensurate ones by restriction of the degrees of freedom. 1 The resulting "hybrid" translational symmetry no longer guarantees wave propagation of elementary excitations by constructive interference. As a consequence, localization lengths and transport properties depend sensitively on various system parameters as well as on the algebraic character of the discommensuration. 2 Recent work exploring this field has concentrated on the continuous one-dimensional (ID) quasiperiodic (qp) Schrodinger equationand its tight-binding version or Poincare map V n + x + \v(n/
The problem of finding the exact energies and configurations for the Frenkel-Kontorova model consisting of particles in one dimension connected to their nearest-neighbors by springs and placed in a periodic potential consisting of segments from parabolas of identical (positive) curvature but arbitrary height and spacing, is reduced to that of minimizing a certain convex function defined on a finite simplex.64.70. Rh, 64.60.Ak, 61.44.+p, 05.45.+b
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