Mario Arioli: December 9, 2010

An introduction to Quantum Graphs and their applications

Mario Arioli,   STFC-RAL, UK

Thursday, December 9, 11:00 a.m. in the CERFACS conference room

Abstract:

In modelling physical phenomena in Physics (quantum wires, photonic crystals), Chemistry (free electron theory of conjugated molecules), Engineering (nanotechnology and microelectronic), we face the problem of solving PDEs defined in very complex domains. In particular, some of these domains are “quasi-one-dimensional” manifolds. Narrow branching media are a typical example: the domain resembles a fattened graph made of nano-tubes linked together. The Quantum Graph approach models these problems by reducing the PDE domain to a proper metric graph G(E,V,?) (E is the set of edges where the measure ? is assigned and V is the set of vertices) and then introducing an Hamiltonian that takes into account the physical properties of the problem and convenient “vertex conditions” that assure that the operators are self-adjoint. We will give a 'naive' introduction to the mathematical and numerical problems connected to Quantum Graphs, and we will highlight some of the problems using simple examples.