Multiscale Analysis in Momentum Space for Quasi-periodic Potential in Dimension Two

TL;DR

This paper proves that a polyharmonic operator with a quasi-periodic potential in two dimensions has a spectrum containing a semi-infinite interval, with eigenfunctions resembling plane waves and Cantor-like isoenergetic curves, using a novel multiscale analysis method.

Contribution

It introduces a new multiscale analysis technique in momentum space to analyze the spectral properties of polyharmonic operators with quasi-periodic potentials.

Findings

Absolutely continuous spectrum contains a semi-axis.

Eigenfunctions approximate plane waves at high energies.

Isoenergetic curves are distorted circles with Cantor set structure.

Abstract

We consider a polyharmonic operator $H=(-\Delta)^l+V(\x)$ in dimension two with $l\geq 2$, $l$ being an integer, and a quasi-periodic potential $V(\x)$. We prove that the absolutely continuous spectrum of $H$ contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves $e^{i< \k,\x>}$ at the high energy region. Second, the isoenergetic curves in the space of momenta $\k$ corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). A new method of multiscale analysis in the momentum space is developed to prove these results.