Schr\"odinger operators on zigzag graphs

TL;DR

This paper analyzes the spectral properties of Schr"odinger operators on zigzag graphs with periodic potentials, revealing the structure of the spectrum, resonances, and inverse spectral mappings, with applications to carbon nanotubes.

Contribution

It provides a detailed spectral analysis, including the Lyapunov function and resonances, and characterizes finite gap potentials and inverse spectral mappings for zigzag graph operators.

Findings

Spectrum consists of absolutely continuous parts and eigenvalues with infinite multiplicity.

All resonances are real and classified into stable and unstable gaps.

Inverse spectral theory describes finite gap potentials and potential-eigenvalue mappings.

Abstract

We consider the Schr\"odinger operator on zigzag graphs with a periodic potential. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe all compactly supported eigenfunctions with the same eigenvalue. We define a Lyapunov function, which is analytic on some Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove that all resonances are real. We determine the asymptotics of the periodic and anti-periodic spectrum and of the resonances at high energy. We show that there exist two types of gaps: i) stable gaps, where the endpoints are periodic and anti-periodic eigenvalues, ii) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function). We obtain the following results from the inverse spectral theory: 1) we describe all finite gap potentials, 2) the mapping: potential -- all eigenvalues is a real analytic isomorphism for some class of potentials. We apply all these results to quasi-1D models of zigzag single-well carbon nanotubes.