Contact interactions on graph superlattices

Pavel Exner

TBD mathscidoc:1910.43029

Journal of Physics A: Mathematical and General, 29, (1), 87, 1996.1
We consider a quantum mechanical particle living on a graph and discuss the behaviour of its wavefunction at graph vertices. In addition to the standard (or-type) boundary conditions with continuous wavefunctions, we investigate two types of a singular coupling which are analogous to the interaction and its symmetrized version for a particle on a line. We show that these couplings can be used to model graph superlattices in which point junctions are replaced by complicated geometric scatterers. We also discuss the band spectra for rectangular lattices with the mentioned couplings. We show that they roughly correspond to their Kronig-Penney analogues: the lattices have bands whose widths are asymptotically bounded and do not approach zero, while the lattice gap widths are bounded. However, if the lattice-spacing ratio is an irrational number badly approximable by rationals, and the coupling constant is small
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@inproceedings{pavel1996contact,
  title={Contact interactions on graph superlattices},
  author={Pavel Exner},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020122257494630558},
  booktitle={Journal of Physics A: Mathematical and General},
  volume={29},
  number={1},
  pages={87},
  year={1996},
}
Pavel Exner. Contact interactions on graph superlattices. 1996. Vol. 29. In Journal of Physics A: Mathematical and General. pp.87. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020122257494630558.
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