Graduate student seminar: Schrodinger Operators and the Kato Square Root Problem.

In this talk I will survey the results of my PhD thesis on the topic of Schr\"{o}dinger operators. The first part of the talk will discuss the question of how one can construct potential dependent averaging operators. The harmonic oscillator potential $V(x) = |x|^{2}$ will be considered as a particular case and a Hardy-Littlewood type maximal operator affiliated with such potential dependent averaging operators operators will be introduced.

 

The second part of the talk will discuss the Kato square root problem for divergence form elliptic operators with potential. On $\mathbb{R}^{n}$, this is

the equivalence statement $\left\Vert\sqrt{V - \mathrm{div} (A \nabla)}u\right\Vert_{L^{2}(\mathbb{R}^{n})} \simeq \left\Vert \nabla u \right\Vert_{L^{2}(\mathbb{R}^{n})} + \left\Vert V^{\frac{1}{2}}u \right\Vert_{L^{2}(\mathbb{R}^{n})}$, where  $A$ is a complex matrix-valued function satisfying an ellipticity condition. I will outline a proof for this problem on $\mathbb{R}^{n}$ for a general potential $V \in L^{1}_{loc}(\mathbb{R}^{n})$. Following this, it will

be shown how this estimate can be used to solve the Kato problem on an arbitrary domain with no regularity assumptions on its boundary, an open problem that has been contested since the original solution to Kato on $\mathbb{R}^{n}$.