Intrinsic Ultracontractivity for a class of Schroedinger Semigroups in $\mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ using Log-Sobolev-inequalities and duality arguments

Abstract

We present a class of potentials $q \colon \mathbb{R}^{n} \to (0,\infty)$ that implies the weighted Schr\"odinger semigroup $\varphi^{-1}\mathrm{e}^{-tH}\varphi$ to map a weighted Lebesgue function space $\mathrm{L}_{\mu}^{1}(\mathbb{R}^{n})$ into a weighted Lebesgue function space $\mathrm{L}_{\mu}^{2}(\mathbb{R}^{n})$ continously at every time $t>0$ by Logarithmic Sobolev inequalities for $H=-\Delta + q(x)$ with it's strictly positive ground state $\varphi \colon \mathbb{R}^{n} \to (0,\infty)$. We use the self-adjointness of $\mathrm{e}^{-tH}$ in $\mathrm{L}^{2}(\mathbb{R}^{n})$ to infer an intrinsic ultracontractivity, i.e. $\forall t>0 \ \exists C_{t} > 0 \ : \ \left| \mathrm{e}^{-tH} u (x) \right| \ \leq \ C_{t} \varphi(x) \| u \|_{2}$ for every $u \in \mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ almost everywhere in $\mathbb{R}^{n}$.