Concentration Properties
In this project we study the concentration phenomena of solutions to the Laplacian eigenvalue (or similar) equation.
These eigenfunctions can be viewed as quantum mechanical states with energy E. The high energy limit is also known as the semiclassical limit. So we expect to see echos of the classical dynamics in the behaviour of large energy states.
We ask where and to what extent do such eigenfunctions (or approximate eigenfunctions) concentrate.
We measure concentration via L^p estimates of the form,
We want to know, for what functions does the inequality hold? Is the inequality sharp?
