Quantum mechanics on graphs

Abstract

We analyse the pmblem of one-dimensional quantum mechanics on arbitrary graphs as idealized models for quantum systems on spaces with non-hivial topologies. In panic& we argue that such models can be made to ac"modate the physical characteristics of wavefunctions on a nehvork of wires and offer several derivations of a panicular junction condition. Throughout we adopt a continuity condition for the wavefunction at each primitive node in the network. Results are applied to the problem of the energy specmm of a system containing one and infinitely many junctions.

Periodic Orbit Theory and Spectral Statistics for Quantum Graphs

“…A finite value of λ i introduces a new length scale. It is natural therefore, to interpret it in physical terms as a representation of a local impurity or an external fields [18,19,27]. We finally note that the above model can be considered as a generalization of the Kronig-Penney model to a multiply connected, yet one dimensional manifold.…”

A single-mode quantum transport in serial-structure geometric scatterers

“…Figure 1: A loop-graph scatterer in a magnetic field only recently as a tool to describe systems of quantum wires -see [2,7,9,10,13,15,16,17,18,20,21,25,27,28,31,34] and references therein. There are also other systems for which graph description could prove to be useful such as objects composed of carbon nanotubes [14,30].…”

Contact interactions on graph superlattices

“…In recent years the interest to quantum mechanics on graphs has been revived -see, e.g., [1,5,6,8,9,16,19,20] and references therein -in particular, as a reaction to the rapid progress of fabrication techniques which allow us nowadays to produce plenty of graph-like structures of a pure semiconductor material, for which graph Hamiltonians represent a natural model. This posed anew the question about physical plausibility of the boundary conditions (1.1).…”