The isospectral fruits of representation theory: quantum graphs and drums

Abstract

Abstract. 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 proved. The complementary part concerns techniques of assembly which are both stated generally and demonstrated. For that purpose, quantum graphs are grist to the mill. We develop the intuition that stands behind the construction as well as the practical skills of producing isospectral objects. We discuss the theoretical implications which include Sunada's theorem of isospectrality [2] arising as a particular case of this method. A gallery of new isospectral examples is presented and some known examples are shown to result from our theory.

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.…”

Schrödinger Operators on Graphs and Geometry II. Spectral Estimates for $${\varvec{L}}_\mathbf{1}$$ L 1 -potentials and an Ambartsumian Theorem

“…Assuming that the edge lengths are rationally independent one may even reconstruct the graph using the trace formula [15,18], but under the condition that the potential is zero. Explicit examples of isospectral graphs have been constructed [4,5,15]. Employing the boundary control method one may reconstruct metric trees [2] without assuming zero potential.…”

Recovering quantum graphs from their Bloch spectrum

“…We define the set of blocks to be the set of equivalence classes. (1) Note that all loops in the graph are blocks, all other blocks are 2-connected.…”

“…, that is a failure of the triangle inequality, we know that B 1 has to be an inner vertex in the block structure of G. The path between the blocks B 2 and B 3 has to pass through B 1 and use some edges within the block B 1 if d(B i , B j ) d(B 1 , B i ) + d(B 1 , B j ) they are attached at the same cut vertex. Within each of these groups the block B 1 is a leaf in the block structure.…”