Can one hear the shape of a graph?

Abstract

We show that the spectrum of the Schrödinger operator on a finite, metric graph determines uniquely the connectivity matrix and the bond lengths, provided that the lengths are non-commensurate and the connectivity is simple (no parallel bonds between vertices and no loops connecting a vertex to itself). That is, one can hear the shape of the graph! We also consider a related inversion problem: A compact graph can be converted into a scattering system by attaching to its vertices leads to infinity. We show that the scattering phase determines uniquely the compact part of the graph, under similar conditions as above.

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

“…In other words (see (13)), the number of nodal domains depend on the quadrant in the (φ 1 , φ 2 ) plane where the eigenvectors point: µ = 1 in the first and the third quadrants, and µ = 2 in the second and the fourth quadrants:…”

“…In [15,16] it was shown that in general, the spectrum does not determine uniquely the length of the bonds and their connectivity. However, it was shown in [13] that quantum graphs whose bond lengths are rationally independent "can be heard" -that is -their spectra determine uniquely their connectivity matrices and their bond lengths. This fact follows from the existence of an exact trace formula for quantum graphs [18,19].…”

New isotope technologies in environmental physics

“…For quantum graphs, existence of pairs of graphs with the same spectrum was proven e.g. in [vBe01,GS01,BPB09]. On the other hand, Gutkin and Smilansky [GS01] also show that for graphs with rationally independent edges, which are strongly coupled (i.e., there are no zeros in the vertex-scattering matrices defined in Section 12) and do not have loops (edges starting and ending at one vertex) and multiple edges between a given pair of the vertices, the spectrum is unique.…”

Resolving isospectral ‘drums’ by counting nodal domains

“…Other pairs of isospectral domains in R 2 were proposed in [13] and discussed further in [14]. Sunada-like quantum graphs were presented in [4]. Milnor's original work on isospectral flat tori in R 16 induced several investigators to find other examples in spaces of lower dimensions.…”