# Recovering quantum graphs from their Bloch spectrum

@article{Rueckriemen2011RecoveringQG, title={Recovering quantum graphs from their Bloch spectrum}, author={Ralf Rueckriemen}, journal={arXiv: Spectral Theory}, year={2011} }

We define the Bloch spectrum of a quantum graph to be the collection of the spectra of a family of Schrodinger operators parametrized by the cohomology of the quantum graph. We show that the Bloch spectrum determines the Albanese torus, the block structure and the planarity of the graph. It determines a geometric dual of a planar graph. This enables us to show that the Bloch spectrum completely determines planar 3-connected quantum graphs.

#### 8 Citations

Inverse Problems for Quantum Graphs: Recent Developments and Perspectives

- Mathematics
- 2011

An introduction into the area of inverse problems for the Schrodinger operators on metric graphs is given. The case of metric finite trees is treated in detail with the focus on matching conditions.… Expand

Stability of eigenvalues of quantum graphs with respect to magnetic perturbation and the nodal count of the eigenfunctions

- Mathematics, Physics
- Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2014

The nth eigenvalue of the Schrödinger operator on a quantum graph is considered as a function of the magnetic perturbation and it is shown that its Morse index at zero magnetic field is equal to φ−(n−1). Expand

Reducible Fermi surfaces for non-symmetric bilayer quantum-graph operators

- Physics, Mathematics
- Journal of Spectral Theory
- 2019

This work constructs a class of non-symmetric periodic Schrodinger operators on metric graphs (quantum graphs) whose Fermi, or Floquet, surface is reducible. The Floquet surface at an energy level is… Expand

Zeros of Eigenfunctions of the Schrodinger Operator on Graphs and Their Relation to the Spectrum of the Magnetic Schrodinger Operator

- Mathematics
- 2014

In this dissertation, we analyze characteristics of eigenfunctions of the Schrödinger operator on graphs. In particular, we are interested in the zeros of the eigenfunctions and their relation to the… Expand

Nodal count of graph eigenfunctions via magnetic perturbation

- Mathematics, Physics
- 2013

We establish a connection between the stability of an eigenvalue under a magnetic perturbation and the number of zeros of the corresponding eigenfunction. Namely, we consider an eigenfunction of… Expand

A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds

- Mathematics
- 2013

In this article, we prove a Lichnerowicz estimate for a compact convex domain of a Kahler manifold whose Ricci curvature satisfies $\Ric \ge k$ for some constant $k>0$. When equality is achieved, the… Expand

Anomalous nodal count and singularities in the dispersion relation of honeycomb graphs

- Mathematics, Physics
- 2015

We study the nodal count of the so-called bi-dendral graphs and show that it exhibits an anomaly: the nodal surplus is never equal to 0 or $\beta$, the first Betti number of the graph. According to… Expand

Reconstruction of an unknown electrical network from their reflectogram by an iterative algorithm based on local identification of peaks and inverse scattering theory

- Engineering, Computer Science
- 2018 IEEE International Instrumentation and Measurement Technology Conference (I2MTC)
- 2018

The method is based upon an iterative algorithm associating the peaks of a reflectogram with unknown scatterers which can be either junction or terminal end of the network, dispelling the ambiguities caused by the complexity of the reflectogram. Expand

#### References

SHOWING 1-10 OF 23 REFERENCES

Trace formulae for quantum graphs

- 2007

Quantum graph models are based on the spectral theory of (differential) Laplace operators on metric graphs. We focus on compact graphs and survey various forms of trace formulae that relate Laplace… Expand

Can one hear the shape of a graph

- Physics, Mathematics
- 2001

We show that the spectrum of the Schroperator on a finite, metric graph determines uniquely the connectivity matrix and the bond lengths, provided that the lengths are non-commensurate and the… Expand

Periodic Orbit Theory and Spectral Statistics for Quantum Graphs

- Physics
- 1998

Abstract We quantize graphs (networks) which consist of a finite number of bonds and vertices. We show that the spectral statistics of fully connected graphs is well reproduced by random matrix… Expand

Jacobian Tori Associated with a Finite Graph and Its Abelian Covering Graphs

- Mathematics, Computer Science
- Adv. Appl. Math.
- 2000

We develop a graph-theoretic analogue of the Jacobian torus, which is defined, for a finite graph X, as the torus H1(X,R)/H1(X,Z) with a natural flat metric. It is observed that this notion,… Expand

Inverse spectral results on even dimensional tori

- Mathematics
- 2008

Given a Hermitian line bundle L over a flat torus M, a connection ∇ on L, and a function Q on M, one associates a Schrodinger operator acting on sections of L; its spectrum is denoted Spec(Q;L,∇).… Expand

Quantum graphs: an introduction and a brief survey

- Physics, Mathematics
- 2008

The purpose of this text is to set up a few basic notions concerning quantum graphs, to indicate some areas addressed in the quantum graph research, and to provide some pointers to the literature.… Expand

First Order Approach and Index Theorems for Discrete and Metric Graphs

- Mathematics, Physics
- 2007

The aim of the present paper is to introduce a unified notion of Laplacians on discrete and metric graphs. In order to cover all self-adjoint vertex conditions for the associated metric graph… Expand

The Isospectral Fruits of Representation Theory: Quantum Graphs and Drums

- Mathematics, Physics
- 2009

We present a method which enables one to construct isospectral objects, such as quantum graphs and drums. One aspect of the method is based on representation theory arguments which are shown and… Expand

Inverse spectral results on two-dimensional tori

- Mathematics
- 1990

is much more complicated than the corresponding problem on surfaces of genus > 2. On one hand, for certain classes of potentials, one can reconstruct V from the spectrum of (1.1); on the other hand,… Expand

Graphs on Surfaces

- Computer Science, Mathematics
- Johns Hopkins series in the mathematical sciences
- 2001

This chapter discusses Embeddings Combinatorially, Contractibility, of Cycles, and the Genus Problem, which focuses on planar graphs and the Jordan Curve Theorem, and colorings of Graphs on Surfaces, which are 5-choosable. Expand