Schrödinger operators on metric graphs with general vertex conditions are studied.Explicit spectral asymptotics is derived in terms of the spectrum of reference Laplacians. A geometric version of Ambartsumian theorem is proven under the assumption that the vertex conditions are asymptotically properly connecting and asymptotically standard. By constructing explicit counterexamples it is shown that the geometric Albartsumian theorem does not hold in general without additional assumptions on the vertex conditions.
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 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.Mathematics Subject Classification. 34L15, 35R30, 81Q10.
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