Douglas Lundholm: Hardy and Lieb-Thirring inequalities for anyonic particle statistics

We consider the many-particle quantum mechanics of anyons, i.e. identical particles in two space dimensions classified by a continuous statistics parameter alpha in [0,1] ranging from bosons (alpha=0) to fermions (alpha=1). These can be modeled by means of completely symmetric (bosonic) wavefunctions with Aharonov-Bohm topological magnetic potentials attached to every particle. We prove a magnetic Hardy inequality for anyons, which in the case that alpha is an odd numerator fraction implies a local exclusion principle for the kinetic energy of such anyons. From this result, and motivated by the original route to stability of ordinary fermionic matter in three dimensions due to Dyson and Lenard, we prove a Lieb-Thirring inequality for these types of anyons. This is recent joint work with Jan Philip Solovej.