Fractional Lieb-Thirring inequality for interacting bosons

The Lieb-Thirring inequality is an elegant combination of
uncertainty and exclusion principles in terms of a semi-classical lower
bound on the kinetic energy of free fermions. Recently, Lundholm, Portmann
and Solovej found an analogue of the Lieb-Thirring inequality for
interacting bosons where the interaction is proportional to the inverse
square of the distance between particles. In this talk I will present a
generalization of their result to fractional kinetic operator. I will also
show that if the corresponding interaction potential is locally
integrable, then the Lieb-Thirring inequality is actually equivalent to a
one-body  Gagliardo-Nirenberg inequality proved recently by Bellazzini,
Frank and Visciglia. The talk is based on joint work with Douglas Lundholm
and Fabian Portmann.