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Letters in Mathematical Physics

On the ground state for quantum graphs

Pavel Kurasov

Published Jul 4, 2019• 18 citations
Publisher

Abstract

Ground-state eigenfunctions of Schrödinger operators can often be chosen positive. We analyse to which extent this is true for quantum graphs—differential operators on metric graphs. It is shown that the theorem holds in the case of generalised delta couplings at the vertices—a new class of vertex conditions introduced in the paper. It is shown that this class of vertex conditions is optimal. Relations to positivity preserving and positivity improving semigroups are clarified.

DOI
10.1007/s11005-019-01192-w

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Ground-state eigenfunctions of Schrödinger operators can often be chosen positive. We analyse to which extent this is true for quantum graphs—differential operators on metric graphs. It is shown that the theorem holds in the case of generalised delta couplings at the vertices—a new class of vertex conditions introduced in the paper. It is shown that this class of vertex conditions is optimal. Relations to positivity preserving and positivity improving semigroups are clarified.
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We consider a collection of bosonic modes corresponding to the vertices of a graph Gamma.\\Gamma.Gamma. Quantum tunneling can occur only along the edges of Gamma\\GammaGamma and a local self-interaction term is present. Quantum entanglement of one vertex with respect the rest of the graph is analyzed in the ground-state of the system as a function of the tunneling amplitude tau.\\tau.tau. The topology of Gamma\\GammaGamma plays a major role in determining the tunneling amplitude tau∗\\tau^*tau∗ which leads to the maximum ground-state entanglement. Whereas in most of the cases one finds the intuitively expected result tau∗=infty\\tau^*=\\inftytau∗=infty we show that it there exists a family of graphs for which the optimal value oftau\\tautau is pushed down to a finite value. We also show that, for complete graphs, our bi-partite entanglement provides useful insights in the analysis of the cross-over between insulating and superfluid ground states
Abstract The purpose of the present paper is to discuss the time dependent Schrödinger equation on a metric graph with time-dependent edge lengths, and the proper way to pose the problem so that the corresponding time evolution is unitary. We show that the well posedness of the Schrödinger equation can be guaranteed by replacing the standard Kirchhoff Laplacian with a magnetic Schrödinger operator with a harmonic potential. We then generalize the result to time dependent families of vertex conditions. We also apply the theory to show the existence of a geometric phase associated with a slowly changing quantum graph.