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Ballistic Transport for Schrödinger Operators with Quasi-periodic Potentials

Yulia Karpeshina, Leonid Parnovski, Roman Shterenberg

Abstract

We prove the existence of ballistic transport for a Schrödinger operator with a generic quasi-periodic potential in any dimension $d>1$.

Ballistic Transport for Schrödinger Operators with Quasi-periodic Potentials

Abstract

We prove the existence of ballistic transport for a Schrödinger operator with a generic quasi-periodic potential in any dimension .

Paper Structure

This paper contains 9 sections, 5 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Prior results on ballistic transport
  3. The Main Result
  4. Spectral Properties of the Operator $H$
  5. Prior results.
  6. Description of the method.
  7. Extension of $\lambda_\infty(\mathbf k)$ from $\mathcal{G}_\infty$ to ${\mathbb{R}}^d$
  8. Extension of $U_{\infty }(\mathbf k,\mathbf x)$ from $\mathcal{G}_\infty$ to ${\mathbb{R}}^d$
  9. Proofs of Proposition \ref{['Prop2']} and Theorem \ref{['Thm1']}

Key Result

Theorem 1.1

For any given set of Fourier coefficients $\{V_{\boldsymbol n}\}$, $V_{-\boldsymbol n}={\bar{V}_{\boldsymbol n}}$, $|\boldsymbol n|\le Q$, $Q\in{\mathbb{N}}$, there exists a subset $\Omega_*=\Omega_*(\{V_{\boldsymbol n}\})\subset [-1/2,1/2]^{dl}$ of basic frequencies with $\hbox{meas}\,(\Omega_*)=1$ there are constants $c_1 = c_1(\Psi_0)>0$ and $T_0 = T_0(\Psi _0)$ such that the solution $\Psi (\m

Theorems & Definitions (9)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 1.3
  • Corollary 1.4
  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 2.4
  • Remark 3.1