PDE Seminar Abstracts

On ${R}^{d}$ we consider the SchrÃ¶dinger operator

$$Lf\left(x\right)=-\Delta f\left(x\right)+V\left(x\right)f\left(x\right),$$

where $\Delta ={\partial}_{{x}_{1}}^{2}+\cdots +{\partial}_{{x}_{d}}^{2}$ and $V\left(x\right)\ge 0$ is a positive function (“potential”).

Let ${T}_{t}=exp\left(-tL\right)$ be the heat semigroup associated with to $L$. In the talk we shall consider the Hardy space

$${H}^{1}\left(L\right):=\left\{f\in {L}^{1}\left({\mathbb{R}}^{d}\right):\underset{t>0}{sup}{T}_{t}f\left(x\right)\in {L}^{1}\left({\mathbb{R}}^{d}\right)\right\}$$

which is a natural substitute of ${L}^{1}\left({\mathbb{R}}^{d}\right)$ in harmonic analysis associated with $L$. Our main interest will be in showing that elements ${H}^{1}\left(L\right)$ have decompositions of the type $f\left(x\right)={\sum}_{k}{\lambda}_{k}{a}_{k}\left(x\right)$, where ${\sum}_{k}\left|{\lambda}_{k}\right|<\infty $ and ${a}_{k}$ (“atoms”) have some nice properties.

In the classical case $V\equiv 0$ on ${\mathbb{R}}^{d}$ an atom is a function $a$ for which there exist a ball $B\subseteq {\mathbb{R}}^{d}$ such that

$$supp\left(a\right)\subseteq B,\phantom{\rule{2em}{0ex}}\parallel a{\parallel}_{\infty}\le |B{|}^{-1},\phantom{\rule{2em}{0ex}}\int a\left(x\right)dx=0.$$

We shall see that for $L=-\Delta +V$ we can still prove some atomic decompositions, but the properties of atoms depend on the dimension d and the potential $V$.

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