Extended States for Polyharmonic Operators with Quasi-periodic Potentials in Dimension Two

TL;DR

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

Contribution

The authors develop a novel multiscale analysis method in momentum space to establish the existence of extended states and Cantor-type isoenergetic curves for polyharmonic operators with quasi-periodic potentials.

Findings

Spectrum contains a semi-axis at high energies.

Eigenfunctions are close to plane waves.

Isoenergetic curves are Cantor-like with holes.

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 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.