Abstract
In this paper we study Schrödinger operators with absolutely integrable potentials on metric graphs. Uniform bounds—i.e. depending only on the graph and the potential—on the difference between the \(n^\mathrm{th}\) eigenvalues of the Laplace and Schrödinger operators are obtained. This in turn allows us to prove an extension of the classical Ambartsumian Theorem which was originally proven for Schrödinger operators with Neumann conditions on an interval. We also extend a previous result relating the spectrum of a Schrödinger operator to the Euler characteristic of the underlying metric graph.
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P. K. was partially supported by the Swedish Research Council (Grant D0497301) and by the Center for Interdisciplinary Research (ZiF) in Bielefeld in the framework of the cooperation group on Discrete and continuous models in the theory of networks.
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Boman, J., Kurasov, P. & Suhr, R. Schrödinger Operators on Graphs and Geometry II. Spectral Estimates for \({\varvec{L}}_\mathbf{1}\)-potentials and an Ambartsumian Theorem. Integr. Equ. Oper. Theory 90, 40 (2018). https://doi.org/10.1007/s00020-018-2467-1
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DOI: https://doi.org/10.1007/s00020-018-2467-1