arXiv subjects

F. Steinbach

Publications and source records attributed to F. Steinbach.

3 recordsLinked to original sources

arXiv2000

Statistics of resonances and of delay times in quasiperiodic Schr"odinger equations

We study the statistical distributions of the resonance widths ${\cal P} (Γ)$, and of delay times ${\cal P} (τ)$ in one dimensional quasi-periodic tight-binding systems with one open channel. Both quantities are found to decay algebraically as $Γ^{-α}$, and $τ^{-γ}$ on small and large scales respectively. The exponents $α$, and $γ$ are related to the fractal dimension $D_0^E$ of the spectrum of the closed system as $α=1+D_0^E$ and $γ=2-D_0^E$. Our results are verified for the Harper model at the metal-insulator transition and for Fibonacci lattices.

cond-mat.mes-hall↗

arXiv2000

New Class of Eigenstates in Generic Hamiltonian Systems

In mixed systems, besides regular and chaotic states, there are states supported by the chaotic region mainly living in the vicinity of the hierarchy of regular islands. We show that the fraction of these hierarchical states scales as $\hbar^{-α}$ and relate the exponent $α=1-1/γ$ to the decay of the classical staying probability $P(t)\sim t^{-γ}$. This is numerically confirmed for the kicked rotor by studying the influence of hierarchical states on eigenfunction and level statistics.

nlin.CD↗

arXiv1998

A covering property of Hofstadter's butterfly

Based on a thorough numerical analysis of the spectrum of Harper's operator, which describes, e.g., an electron on a two-dimensional lattice subjected to a magnetic field perpendicular to the lattice plane, we make the following conjecture: For any value of the incommensurability parameter sigma of the operator its spectrum can be covered by the bands of the spectrum for every rational approximant of sigma after stretching them by factors with a common upper bound. We show that this conjecture has the following important consequences: For all irrational values of sigma the spectrum is (i) a zero measure Cantor set and has (ii) a Hausdorff dimension less or equal to 1/2. We propose that our numerical approach may be a guide in finding a rigorous proof of these results.

cond-mat.mes-hall↗