Optimal Potentials for Quantum Graphs
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Schrödinger operators on metric graphs with delta couplings at the vertices are studied. We discuss which potential and which distribution of delta couplings on a given graph maximise the ground state energy, provided the integral of the potential and the sum of strengths of the delta couplings are fixed. It appears that the optimal potential if it exists is a constant function on its support formed by a set of intervals separated from the vertices. In the case where the optimal configuration does not exist explicit optimising sequences are presented.
Mathematics Subject Classification35P15 81Q35
We wish to thank Gregory Berkolaiko, James Kennedy and Delio Mugnolo for the discussion regarding the application of the cutting principles. The authors were partially supported by the Center for Interdisciplinary Research (ZiF) in Bielefeld in the framework of the cooperation group Discrete and continuous models in the theory of networks. P. K. was partially supported by the Swedish Research Council Grant D0497301. We would like to thank anonymous referees for careful reading of the manuscript and constructive suggestions.
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