PDE-seminar Speaker: Søren Fournais, Aarhus university

ABSTRACT

In this talk we will discuss recent work on Hardy-Lieb-Thirring inequalities for the Pauli operator. The classical Lieb-Thirring inequality estimates the sum of the negative eigenvalues of a Schr\”{o}dinger operator 
$-\Delta + V$ by an integral of a power of the potential. In $3$-dimensions, this becomes
$$
\tr( - \Delta + V)_{-} \leq C \int (V(x))_{-}^{5/2}\,dx
$$
The classical Hardy inequality states that (also in $3D$),
$$
-Delta - \frac{1}{4 |x|^2} \geq 0,
$$
where the constant $\frac{1}{4}$ is the sharp constant for this bound.

It is well known, that these inequalities can be combined to yield "Hardy-Lieb-Thirring inequalities”, i.e. the Lieb-Thirring inequality above still holds (possibly with a different constant) if $V$ is replaced by $- \frac{1}{4 |x|^2} + V$ on the left side. 

In this talk we will discuss similar inequalities, where the non-relativistic kinetic energy operator $-\Delta$ is replaced by a magnetic Pauli operator. In particular, we will discuss a relativistic version, where the kinetic energy is the square root of a Pauli operator, and where $ \frac{1}{4 |x|^2}$ is replaced by $\frac{c_H}{|x|}$, with $c_H$ being the critical Hardy constant for the relativistic problem.

This is joint work with Gonzalo Bley.