Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Level spacing statistics for the multi-dimensional quantum harmonic oscillator: algebraic case

Alan Haynes, Roland Roeder

Abstract

We study the statistical properties of the spacings between neighboring energy levels for the multi-dimensional quantum harmonic oscillator that occur in a window $[E,E+ΔE)$ of fixed width $ΔE$ as $E$ tends to infinity. This regime provides a notable exception to the Berry-Tabor Conjecture from Quantum Chaos and, for that reason, it was studied extensively by Berry and Tabor in their seminal paper from 1977. We focus entirely on the case that the (ratios of) frequencies $ω_1,ω_2,\ldots,ω_d$ together with $1$ form a basis for an algebraic number field $Φ$ of degree $d+1$, allowing us to use tools from algebraic number theory. This special case was studied by Dyson, Bleher, Bleher-Homma-Ji-Roeder-Shen, and others. Under a suitable rescaling, we prove that the distribution of spacings behaves asymptotically quasiperiodically in $\log E$. We also prove that the distribution of ratios of neighboring spacings behaves asymptotically quasiperiodically in $\log E$. The same holds for the distribution of finite words in the finite alphabet of rescaled spacings. Mathematically, our work is a higher dimensional version of the Steinhaus Conjecture (Three Gap Theorem) involving the fractional parts of a linear form in more than one variable, and it is of independent interest from this perspective.

Level spacing statistics for the multi-dimensional quantum harmonic oscillator: algebraic case

Abstract

We study the statistical properties of the spacings between neighboring energy levels for the multi-dimensional quantum harmonic oscillator that occur in a window of fixed width as tends to infinity. This regime provides a notable exception to the Berry-Tabor Conjecture from Quantum Chaos and, for that reason, it was studied extensively by Berry and Tabor in their seminal paper from 1977. We focus entirely on the case that the (ratios of) frequencies together with form a basis for an algebraic number field of degree , allowing us to use tools from algebraic number theory. This special case was studied by Dyson, Bleher, Bleher-Homma-Ji-Roeder-Shen, and others. Under a suitable rescaling, we prove that the distribution of spacings behaves asymptotically quasiperiodically in . We also prove that the distribution of ratios of neighboring spacings behaves asymptotically quasiperiodically in . The same holds for the distribution of finite words in the finite alphabet of rescaled spacings. Mathematically, our work is a higher dimensional version of the Steinhaus Conjecture (Three Gap Theorem) involving the fractional parts of a linear form in more than one variable, and it is of independent interest from this perspective.

Paper Structure

This paper contains 14 sections, 8 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Physical Perspective
  3. Mathematical Perspective
  4. Plan for the paper
  5. Acknowledgments
  6. Notation and preliminary results
  7. Proof of Part (1) of Theorem A' (Uniform Labeling)
  8. Proof of Part (2) of Theorem A' (Quasiperiodicity)
  9. Partitions of $M(t)$ and $R$.
  10. Matrix representation of multiplication by elements of $\Phi$ and their logarithms.
  11. Reduction to analysis of normalized expansions of $u_1(t)$
  12. Analysis of normalized expansions of $u_1(t)$ as $t \rightarrow \infty$.
  13. Quasiperiodicity of finite words and Proof of Theorem B'
  14. Worked example: a totally real cubic field

Key Result

Theorem \oldthetheorem

BlehHommJiRoedShen2012 If $1,\omega_1,\ldots ,\omega_d\in\mathbb{R}$ form a $\mathbb{Q}$-basis for an algebraic number field $\Phi$, then there exists a finite set such that every spacing $\Delta_i$ has the form $us_j$ for some unit $u$ in the ring of integers $\mathbb{Z}_\Phi$ and some $s_j\in \mathcal{S}$.

Theorems & Definitions (21)

  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Proposition \oldthetheorem: Prop. 5.1 from BlehHommJiRoedShen2012
  • Remark \oldthetheorem
  • Proposition \oldthetheorem: Prop. 5.2 from BlehHommJiRoedShen2012
  • Proposition \oldthetheorem: Prop. 5.3 from BlehHommJiRoedShen2012
  • ...and 11 more