Spectra of Schroedinger operators on equilateral quantum graphs

TL;DR

This paper analyzes the spectra of magnetic Schrödinger operators on equilateral quantum graphs, revealing a relationship with discrete magnetic Laplacians and Hill operators, and providing spectral gap estimates independent of graph structure.

Contribution

It establishes a novel spectral correspondence between quantum graphs and Hill operators, enabling gap estimates regardless of graph topology.

Findings

Spectrum is the preimage of the combinatorial spectrum under an entire function.

Number of spectral gaps can be estimated from below using Hill operator properties.

Spectral analysis reduces to discrete and Hill operator problems.

Abstract

We consider magnetic Schroedinger operators on quantum graphs with identical edges. The spectral problem for the quantum graph is reduced to the discrete magnetic Laplacian on the underlying combinatorial graph and a certain Hill operator. In particular, it is shown that the spectrum on the quantum graph is the preimage of the combinatorial spectrum under a certain entire function. Using this correspondence we show that that the number of gaps in the spectrum of the Schroedinger operators admits an estimate from below in terms of the Hill operator independently of the graph structure.