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Quantum graphs which sound the same. (English) Zbl 1132.37312
Khanna, Faqir (ed.) et al., Nonlinear dynamics and fundamental interactions. Proceedings of the NATO Advanced Research Workshop, Tashkent, Uzbekistan, October 10–16, 2004. Dordrecht: Springer (ISBN 1-4020-3948-4/pk). NATO Science Series II: Mathematics, Physics and Chemistry 213, 17-30 (2006).
Summary: We present a method to construct pairs of isospectral quantum graphs which are not isometric. These graphs are the analogues of the family of isospectral domains in \(\mathbb{R}^2\) which were first introduced by C. Gordon, D. Webb and S. Wolpert [Invent. Math. 110, No. 1, 1–22 (1992; Zbl 0778.58068)], recently enlarged by P. Buser, J. Conway, P. Doyle and K.-D. Semmler [Int. Math. Res. Not. 1994, No. 9, 391–400 (1994; Zbl 0837.58033)], and discussed further by Y. Okada and A. Shudo [J. Phys. A, Math. Gen. 34, No. 30, 5911–5922 (2001; Zbl 0986.58017)].
For the entire collection see [Zbl 1120.81004].

37E25 Dynamical systems involving maps of trees and graphs
58J53 Isospectrality