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On the hot spots of quantum graphs

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  • We undertake a systematic investigation of the maxima and minima of the eigenfunctions associated with the first nontrivial eigenvalue of the Laplacian on a metric graph equipped with standard (continuity–Kirchhoff) vertex conditions. This is inspired by the famous hot spots conjecture for the Laplacian on a Euclidean domain, and the points on the graph where maxima and minima are achieved represent the generically "hottest" and "coldest" spots of the graph. We prove results on both the number and location of the hot spots of a metric graph, and also present a large number of examples, many of which run contrary to what one might naïvely expect. Amongst other results we prove the following: (i) generically, up to arbitrarily small perturbations of the graph, the points where minimum and maximum, respectively, are attained are unique; (ii) the minima and maxima can only be located at the vertices of degree one or inside the doubly connected part of the metric graph; and (iii) for any fixed graph topology, for some choices of edge lengths all minima and maxima will occur only at degree-one vertices, while for others they will only occur in the doubly connected part of the graph.

    Mathematics Subject Classification: Primary: 34B45; Secondary: 34L10, 35R02, 81Q35.

    Citation:

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  • Figure 3.1.  Left: a path graph with the hot spots marked in grey. Right: a cycle graph; here every point is a hot spot

    Figure 3.2.  A pumpkin and a star graph. The set $ M (\Gamma) $ in the equilateral case is marked in grey

    Figure 3.3.  A flower graph and a complete graph. The hot spots for the equilateral case are marked in grey

    Figure 3.4.  A lasso graph with its hot spots in grey

    Figure 4.1.  The perturbed "figure-8" graph of Example 4.7 and its hot spots in grey

    Figure 4.2.  The perturbed path graph of Example 4.8 and its hot spots in grey

    Figure 5.1.  The complete graph admits an eigenfunction for $ \mu_2 $ whose maximum is at $ v_0 $ and minimum is achieved at the other $ v_k $, with no other critical points

    Figure 6.1.  The star graph $ \Gamma^* $, the intermediate tree $ \Gamma^\top $, and the final, symmetric tree $ \Gamma $ in the case $ m = 5 $

    Figure 7.1.  The discrete graph $ {{\mathcal G}} $, a "pumpkin on a stick"

    Figure 7.2.  Metric graph incarnations of the discrete graph $ {{\mathcal G}} $ from Figure 7.1 together with their hot spots (grey)

    Figure 8.1.  A graph for which we expect that $ M = \{v_-, v_2, v_3\} $ is possible even if the edge lengths are incommensurable

    Figure 8.2.  A candidate graph for Conjecture 8.9. Candidate locations for the hot spots are marked in grey (the precise location will depend on the respective edge lengths and the "thickness" of the pumpkins, and could be within the pumpkins)

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