# 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 Schrodinger 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  discuss the history and context of the Kato square root problem and outline a proof of the Kato estimate with potential for a large class of potentials that includes $L^{n/2}$ and the reverse Holder class $RH_{n/2}$.