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 $[E,E+\Delta E)$ of fixed width $\Delta E$ 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 $\omega_1,\omega_2,\ldots,\omega_d$ together with $1$ form a basis for an algebraic number field $\Phi$ 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.