Douglas Lundholm: Hardy and Lieb-Thirring inequalities for anyons

We consider the many-particle quantum mechanics of anyons, i.e. identical particles in two space dimensions with a continuous statistics parameter alpha in [0,1] ranging from bosons (alpha = 0) to fermions (alpha = 1). These can be modeled using completely symmetric wavefunctions and 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 Dyson and Lenard's original approach to the stability of ordinary fermionic matter in three dimensions, follows also a Lieb-Thirring inequality for these types of anyons. This is joint work with Jan Philip Solovej.