# Quasi-isospectrality on Quantum Graphs

@article{Rueckriemen2012QuasiisospectralityOQ, title={Quasi-isospectrality on Quantum Graphs}, author={Ralf Rueckriemen}, journal={The Journal of Geometric Analysis}, year={2012}, volume={25}, pages={306-316} }

Consider two quantum graphs with the standard Laplace operator and non-Robin type boundary conditions at all vertices. We show that if their eigenvalue-spectra agree everywhere aside from a sufficiently sparse set, then the eigenvalue-spectra and the length-spectra of the two quantum graphs are completely identical. Similarly, if their length-spectra agree everywhere aside from a sufficiently sparse set, then the quantum graphs have the same eigenvalue-spectrum and length-spectrum.

#### 2 Citations

Asymptotically isospectral quantum graphs and generalised trigonometric polynomials

- Mathematics
- 2020

Abstract The theory of almost periodic functions is used to investigate spectral properties of Schrodinger operators on metric graphs, also known as quantum graphs. In particular we prove that two… Expand

The nodal count {0,1,2,3,…} implies the graph is a tree

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

It is shown that it is not possible for all (or even almost all, in the metric case) the eigenvalues to exhibit a diamagnetic behaviour, and the converse theorems for both discrete and metric graphs are proved. Expand

#### References

SHOWING 1-10 OF 25 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

The Trace Formula for Quantum Graphs with General Self Adjoint Boundary Conditions

- Mathematics, Physics
- 2008

Abstract.We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of… 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

Quantum graphs: Applications to quantum chaos and universal spectral statistics

- Physics
- 2006

During the last few years quantum graphs have become a paradigm of quantum chaos with applications from spectral statistics to chaotic scattering and wavefunction statistics. In the first part of… Expand

Quantum Wires with Magnetic Fluxes

- Mathematics, Physics
- 2003

Abstract: In the present article magnetic Laplacians on a graph are analyzed. We provide a complete description of the set of all operators which can be obtained from a given self-adjoint Laplacian… Expand

Heat kernels on metric graphs and a Trace Formula

- Mathematics, Physics
- 2007

We study heat semigroups generated by self-adjoint Laplace operators on metric graphs characterized by the property that the local scattering matrices associated with each vertex of the graph are… Expand

On a Spectral Analog of the Strong Multiplicity One Theorem

- Mathematics
- 2010

We prove spectral analogs of the classical strong multiplicity one theorem for newforms. Let Γ 1 and Γ 2 be uniform lattices in a semisimple group G. Suppose all but finitely many irreducible unitary… 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

Nodal domains on isospectral quantum graphs: the resolution of isospectrality ?

- Mathematics, Physics
- 2006

We present and discuss isospectral quantum graphs which are not isometric. These graphs are the analogues of the isospectral domains in which were introduced recently in Gordon et al (1992 Bull. Am.… Expand

A refinement of strong multiplicity one for spectra of hyperbolic manifolds

- Mathematics
- 2011

Let $\calM_1$ and $\calM_2$ denote two compact hyperbolic manifolds. Assume that the multiplicities of eigenvalues of the Laplacian acting on $L^2(\calM_1)$ and $L^2(\calM_2)$ (respectively,… Expand