We study a family of closed quantum graphs described by one singular vertex of order n = 4. By suitable choice of the parameters specifying the singular vertex, we can construct a closed path in the parameter space that physically corresponds to the smooth interpolation of different topologies - a ring, separate two lines, separate two rings, two rings with a contact point. We find that the spectrum of a quantum particle on this family of graphs shows quantum holonomy.
We discuss approximations of the vertex coupling on a star-shaped quantum graph of n edges in the singular case when the wave functions are not continuous at the vertex and no edge-permutation symmetry is present. It is shown that the Cheon–Shigehara technique using δ interactions with nonlinearly scaled couplings yields a 2n-parameter family of boundary conditions in the sense of norm resolvent topology. Moreover, using graphs with additional edges, one can approximate the [Formula: see text]-parameter family of all time-reversal invariant couplings.
EXAMPLES OF QUANTUM HOLONOMY WITH TOPOLOGY CHANGES
