Scattering theory on graphs (2): the Friedel sum rule

Details Scattering theory on graphs (2): the Friedel sum rule Scattering theory on graphs (2): the Friedel sum rule

2001-12-12

arxiv.org/abs/cond-mat/0112225v2

J.Phys.A: Math.Gen.35 (2002) 3389-3407

Physics

Condensed Matter

Mesoscale and Nanoscale Physics

20 pages, LaTeX, 6 eps figures

Scientific paper

10.1088/0305-4470/35/15/303

We consider the Friedel sum rule in the context of the scattering theory for the Schr\"odinger operator $-\Dc_x^2+V(x)$ on graphs made of one-dimensional wires connected to external leads. We generalize the Smith formula for graphs. We give several examples of graphs where the state counting method given by the Friedel sum rule is not working. The reason for the failure of the Friedel sum rule to count the states is the existence of states localized in the graph and not coupled to the leads, which occurs if the spectrum is degenerate and the number of leads too small.

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