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"A Hardy-Lieb-Thirring inequality for fractional Pauli operators"

Seminarium

Tid: 2018-03-14 15:30 till: 16:30
Plats:MH:333
Kontakt:mikael.persson_sundqvist@math.lth.se


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.