We reconsider a one-dimensional scattering problem in the Schrödinger equation with a Cantor-like potential. The reflection coefficient obeys a scaling law for sufficiently large wave number . The scaling law is expressed with a universal function characterized by a multifractal.
DOI:https://doi.org/10.1103/PhysRevE.95.032214
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(a) Reflection coefficient by a potential wall located at and 0.32 for . (b) A schematic figure of an array of potential walls. (c) A periodic array of potential walls. (d) Reflection coefficient by an array of potential walls of , and .
Reuse & PermissionsReflection coefficients as a function of for (a) the regular pre-Cantor potential of the six stage and (b) a perturbed pre-Cantor potential of the six stage where the width of each potential wall is randomly distributed between and . A semilogarithmic plot is used for .
Reuse & Permissions(a) Reflection coefficients for (solid curve) and 10 (dashed curve) at . (b) Reflection coefficients for (solid curve) and 10 (dashed curve) at .
Reuse & Permissions(a) Reflection coefficients for at (solid curve) and 20 (dashed curve) in a double-logarithmic scale. The dash-dotted line denotes . (b) at (solid curve) and 20 (dashed curve) and (dotted line) for .
Reuse & Permissions(a) in a semilogarithmic plot of . Periodicity is observed in the logarithmic scale of . (b) for in a normal scale. (c) Comparison of three curves: vs (dotted curve) for vs (dashed curve) for , and vs (solid curve) for .
Reuse & Permissions(a) for at and 2. (b) Relationship between and . (c) Generalized dimension (solid curve) and (dashed curve). (d) spectrum.
Reuse & Permissions
