The Spectrum of a Schr\"odinger Operator With Small Quasi-Periodic Potential is Homogeneous

TL;DR

This paper proves that for a small quasi-periodic potential with Diophantine frequency, the spectrum of the Schrödinger operator is homogeneous, extending understanding of spectral properties in quantum mechanics.

Contribution

It establishes the homogeneity of the spectrum for small quasi-periodic Schrödinger operators with Diophantine frequencies, a novel result in spectral theory.

Findings

Spectrum is homogeneous in the Carleson sense

Homogeneity holds for small exponentially decaying potentials

Results apply to operators with Diophantine frequency vectors

Abstract

We consider the quasi-periodic Schr\"odinger operator $$ [H \psi](x) = -\psi"(x) + V(x) \psi(x) $$ in $L^2(\mathbb{R})$, where the potential is given by $$ V(x) = \sum_{m \in \mathbb{Z}^\nu \setminus \{ 0 \}} c(m)\exp (2\pi i m \omega x) $$ with a Diophantine frequency vector $\omega = (\omega_1, \dots, \omega_\nu) \in \mathbb{R}^\nu$ and exponentially decaying Fourier coefficients $|c(m)| \le \varepsilon \exp(-\kappa_0|m|)$. In the regime of small $\varepsilon > 0$ we show that the spectrum of the operator $H$ is homogeneous in the sense of Carleson.