 95415 Exner P.
 Contact interactions on graph superlattices
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Sep 7, 95

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Abstract. 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 delta type) boundary conditions
with continuous wavefunctions, we investigate two types of a singular
coupling which are analogous to the delta' interaction and its
symmetrized version for 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
KronigPenney analogues: the delta' lattices have bands whose
widths are asymptotically bounded and do not approach zero, while the
delta lattice gap widths are bounded. However, if the
latticespacing ratio is an irrational number badly approximable by
rationals, and the delta coupling constant is small enough,
the delta lattice has no ggaps above the threshold of the
spectrum. On the other hand, infinitely many gaps emerge above a
critical value of the coupling constant; for almost all ratios this
value is zero.
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