Viktor Qvarfordt: Quantum Snowflake

The Schrödinger equation is solved on the quantum snowflake graph in order to determine the graph's scattering properties, i.e. to characterize reflection and absorption for waves of varying energies that are sent into the snowflake graph. The quantum snowflake is defined as a quantum graph that in addition is a self-similar tree.

By exploiting the self-similarity and symmetries of the graph a method is developed for reducing the snowflake graph to a line graph with special matching conditions. On this line graph the complexity is further reduced by combining vertices into one meta-vertex. This procedure greatly simplifies the study of the quantum snowflake and gives a better understanding of the internal scattering.

It is showed that the periodic snowflake graph admits band gap structure similar to crystalline materials. A recursive expression for the scattering coefficient for the general snowflake graph with n generations is found.