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Quantum Graphs which Optimize the Spectral Gap

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Abstract

A finite discrete graph is turned into a quantum (metric) graph once a finite length is assigned to each edge and the one-dimensional Laplacian is taken to be the operator. We study the dependence of the spectral gap (the first positive Laplacian eigenvalue) on the choice of edge lengths. In particular, starting from a certain discrete graph, we seek the quantum graph for which an optimal (either maximal or minimal) spectral gap is obtained. We fully solve the minimization problem for all graphs. We develop tools for investigating the maximization problem and solve it for some families of graphs.

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Authors and Affiliations

  1. Department of Mathematics, Technion–Israel Institute of Technology, 32000, Haifa, Israel

    Ram Band

  2. Laboratoire Jacques-Louis Lions, UMR 7598, Université Pierre et Marie Curie, 75252, Paris Cedex 05, France

    Guillaume Lévy

Authors
  1. Ram Band
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  2. Guillaume Lévy
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Corresponding author

Correspondence to Ram Band.

Additional information

Communicated by Jan Derezinski.

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Band, R., Lévy, G. Quantum Graphs which Optimize the Spectral Gap. Ann. Henri Poincaré 18, 3269–3323 (2017). https://doi.org/10.1007/s00023-017-0601-2

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  • DOI: https://doi.org/10.1007/s00023-017-0601-2

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