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Continuum Fibonacci Schrödinger Operators in the Strongly Coupled Regime

David Damanik, Mark Embree, Jake Fillman, Anton Gorodetski, May Mei

Abstract

We study Schrödinger operators on the real line whose potentials are generated by the Fibonacci substitution sequence and a rule that replaces symbols by compactly supported potential pieces. We consider the case in which one of those pieces is identically zero, and study the dimension of the spectrum in the large-coupling regime. Our results include a generalization of theorems regarding explicit examples that were studied previously and a counterexample that shows that the naïve generalization of previously established statements is false. In particular, in the aperiodic case, the local Hausdorff dimension of the spectrum does not necessarily converge to zero uniformly on compact subsets as the coupling constant is sent to infinity.

Continuum Fibonacci Schrödinger Operators in the Strongly Coupled Regime

Abstract

We study Schrödinger operators on the real line whose potentials are generated by the Fibonacci substitution sequence and a rule that replaces symbols by compactly supported potential pieces. We consider the case in which one of those pieces is identically zero, and study the dimension of the spectrum in the large-coupling regime. Our results include a generalization of theorems regarding explicit examples that were studied previously and a counterexample that shows that the naïve generalization of previously established statements is false. In particular, in the aperiodic case, the local Hausdorff dimension of the spectrum does not necessarily converge to zero uniformly on compact subsets as the coupling constant is sent to infinity.

Paper Structure

This paper contains 12 sections, 19 theorems, 129 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main Goals
  3. Results
  4. Background
  5. Subshifts
  6. The Fibonacci Subshift
  7. Trace Map, Invariant, and Local Dimension of the Spectrum
  8. Proofs of Main Results
  9. A Nonlinear Eigenvalue Formulation
  10. A Very Brief Review of Floquet Theory
  11. A Nonlinear Eigenvalue Problem for Periodic Approximations
  12. Singular Vectors of Hyperbolic Matrices

Key Result

Theorem 1.1

There exist $f_0 \neq f_1 \in C_0([0,1])$, $E \in {\mathbb R}$, and $\lambda_k \uparrow \infty$ such that

Figures (4)

  • Figure 1: The lower portion of the spectrum for periodic approximations (period $p=13$) for two continuum Fibonacci operators with $f_0 = 0\cdot \chi_{[0,1)}$, with $f_1$ constant (left) and a $C^\infty$ bump function that is positive for $x\in(0,1)$ (right). For each value of $\lambda$, the corresponding spectrum is a horizontal slice of the plot.
  • Figure 2: For the piecewise constant potential with $c=4$ (shown on the top), the bottom plots show a portion of the spectrum for periodic approximations ($p=F_4=5$, left; $p=F_6=13$, right). The vertical red lines indicate the values of $E=4\pi^2$ and $E=16\pi^2$.
  • Figure 3: For the potential in Figure \ref{['fig:prop32a']}, now focusing on the spectrum near $E=4\pi^2$ (vertical red line) and the first $\lambda>E$ for which ${\rm Tr}\ \mathbf{B}(E, \lambda)=0$ (horizontal red line), for periodic approximations ($p=F_4=5$, left; $p=F_6=13$, right). We observe $F_3=3$ and $F_5=8$ bands in these plots.
  • Figure 4: For a smooth potential involving a split function $f_1$ (shown on the top), the bottom plots show a portion of the spectrum for the periodic approximation with $p=F_4=5$ (left), and a zoom for $p=F_6=13$ (right) showing a confluence of $F_5=8$ spectral bands around $E=4\pi^2$ (vertical red line).

Theorems & Definitions (41)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 2.1: DFG DFG2014
  • Proposition 2.2: DFG DFG2014
  • Proposition 2.3: DFG DFG2014
  • Theorem 2.4: DFG DFG2014
  • Proposition 2.5
  • proof
  • ...and 31 more