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The isospectral fruits of representation theory: quantum graphs and drums. (English) Zbl 1176.58019
A quantum graph is a graph \(\Gamma\) whose edges are represented as segments with a specified length, each carrying the ordinary Laplacian \(d^2/dx^2\) subject to boundary conditions at the vertices. If a symmetry group \(G\) acts on this graph, then for representations of \(G\) or of subgroups \(H_i\), the Laplacian can be restricted to functions that transform according to these representations, or, equivalently, a quotient graph modulo such representations, \(\Gamma/R\), can be constructed on which the functions are defined.
A representation \(R\) of a subgroup \(H\) algebraically induces a representation \(\text{Ind}_H^GR\) (of higher dimension) for the full group \(G\), and \(\Gamma/R\) has the same Laplace spectrum as \(\Gamma/\text{Ind}_H^GR\). In particular, if two representations \(R_i\) of subgroups \(H_i\) induce the same representation, then the \(\Gamma/R_i\) are isospectral. Actually, the quotient graph \(\Gamma/\text{Ind}_H^GR\) is defined in terms of not just the abstract representation, but also of some basis in the vector space on which the representation acts. This produces a wide variety of examples for isospectral quantum graphs.
The authors point out that the main reason for using quantum graphs is the general availability of quotients, but that the method generalizes to other settings (e.g., manifolds or domains). Sunada’s construction principle arises as a special case: subgroups are almost conjugate if and only if their trivial representations induce the same representation on the full group. The authors explain the most popular examples as instances of their general representation theoretic framework, and they also discuss transplantation of eigenfunctions in this context. The paper is written in excellent expository style.

58J53 Isospectrality
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
20C15 Ordinary representations and characters
34B45 Boundary value problems on graphs and networks for ordinary differential equations
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58C40 Spectral theory; eigenvalue problems on manifolds
35P05 General topics in linear spectral theory for PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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