Local Friedel sum rule on graphs

Abstract

We consider graphs made of one-dimensional wires connected at vertices and on which may live a scalar potential. We are interested in a scattering situation where the graph is connected to infinite leads. We investigate relations between the scattering matrix and the continuous part of the local density of states, the injectivities, emissivities and partial local density of states. Those latter quantities can be obtained by attaching an extra lead at the point of interest and by investigating the transport in the limit of zero transmission into the additional lead. In addition to the continuous part related to the scattering states, the spectrum of graphs may present a discrete part related to states that remain uncoupled to the external leads. The theory is illustrated with the help of a few simple examples.

25 pages, LaTeX, 8 figures

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Cited by in corpus (12)

• Interference traps waves in open system: Bound states in the continuum
• Wigner time delay and related concepts -- Application to transport in coherent conductors
• Wigner time-delay distribution in chaotic cavities and freezing transition
• Functionals of the Brownian motion, localization and metric graphs
• Bound states in the continuum in open Aharonov-Bohm rings
• Statistical distribution of the Wigner-Smith time-delay matrix moments for chaotic cavities
• Wigner-Smith time-delay matrix in chaotic cavities with non-ideal contacts
• Direct computation of scattering matrices for general quantum graphs
• Charge and currents distribution in graphs
• Wigner-Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity
• $\zeta$-regularised spectral determinants on metric graphs
• Injectance and a paradox