We present a method for constructing families of isospectral systems, using linear representations of finite groups. We focus on quantum graphs, for which we give a complete treatment. However, the method presented can be applied to other systems such as manifolds and two-dimensional drums. This is demonstrated by reproducing some known isospectral drums, and new examples are obtained as well. In particular, Sunada's method (Ann. Math. 121, 169-186, 1985) is a special case of the one presented.
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Acknowledgements It is an honor to acknowledge U. Smilansky, who is the initiator of this work and an enthusiastic promoter of it, and a pleasure to thank Z. Sela for his support and encouragement. We are grateful to M. Sieber for sharing with us his notes, which led to the construction of the isospectral pair of dihedral graphs. We are indebted to I. Yaakov whose wise remark has led us to examine inductions of representations. It is a pleasure to acknowledge G. Ben-Shach for the fruitful discussions which promoted the research. We thank D. Schüth for the patient examination of the work and the generous support, and P. Kuchment for his essential help with analysis over quantum graphs. The comments and suggestions offered by J. Brüning, L. Friedlander, S. Gnutzmann, O. Post, Z. Rudnick, and M. Solomyak are highly appreciated. The work was supported by the Minerva Center for non-linear Physics and the Einstein (Minerva) Center at the Weizmann Institute, by an ISF fellowship, and by grants from the GIF (grant I-808-228.14/2003), and BSF (grant 2006065).
- Boundary value problem
- Hear the shape of a drum
- Linear representations
- Quantum graphs