Hardy spaces and Schrödinger operators

Abstract

On $R^d$ we consider the Schrödinger operator

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

which is a natural substitute of $L^1\left({ℝ}^d\right)$ 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 ${ℝ}^d$ an atom is a function $a$ for which there exist a ball $B\subseteq {ℝ}^d$ such that

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 $V$.