Quantum Graphs II: Some spectral properties of quantum and combinatorial graphs

TL;DR

This paper explores spectral properties of infinite quantum and combinatorial graphs, introducing theorems on spectrum detection, spectral gap opening, and the nature of eigenfunctions, with implications for physics and mathematics.

Contribution

It proves a Schnol type theorem for spectrum detection, establishes a spectral gap theorem for decorated quantum graphs, and characterizes eigenfunctions in periodic graphs.

Findings

A Schnol type theorem for spectrum detection with subexponential growth eigenfunctions.

Spectral gap opening for decorated quantum graphs.

Periodic graphs with point spectrum have eigenfunctions with compact support.

Abstract

The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and other areas. A Schnol type theorem is proven that allows one to detect that a point belongs to the spectrum when a generalized eigenfunction with an subexponential growth integral estimate is available. A theorem on spectral gap opening for ``decorated'' quantum graphs is established (its analog is known for the combinatorial case). It is also shown that if a periodic combinatorial or quantum graph has a point spectrum, it is generated by compactly supported eigenfunctions (``scars'').