5. Theoretical Physics (CSN4)

Squeezing networks to graphs

by Dr Pavel EXNER (Nuclear Physics Inst./Doppler Inst.Math Physics)

Aula Bruno Touschek (LNF)

Aula Bruno Touschek


Via Enrico Fermi, 40 00044 Frascati (Roma)
The talk is devoted to the problem to which Gianfausto Dell’Antonio made a significant contri- bution, namely approximation of quantum graphs by ’fat graphs’ or tubular networks. It is now well known that the answer depends substantially on the boundary conditions imposed at the network. While Gianfausto’s result concern mostly the Dirichlet case, my aim here is to present a general solution in the Neumann one. It is based on two recent ideas. The first comes from a common work with Olaf Post: it will be shown that using suitably scaled Schr ̈odinger opera- tors one can approximate vertex couplings beyond the Kirchhoff case, including those with wave functions discontinuous at the vertices. Next I will describe another result obtained together with Taksu Cheon and Ondˇrej Turek on approximations by Schr ̈odinger operators on graphs which shows a way how the graph prob- lem can be solved in full generality. Combining the two techniques, on can approximate any coupling using families of scaled Schr ̈odinger operators on Neumann networks.dy [9] Efimov effects - what about the mysterious ”unitary Bose gas” ?