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 Schrödinger semigroup $\varphi^{-1}\mathrm{e}^{-tH}\varphi$ to map a weighted Lebesgue function space $\mathrm{L}_μ^{1}(\mathbb{R}^{n})$ into a weighted Lebesgue function space $\mathrm{L}_μ^{2}(\mathbb{R}^{n})$ continously at every time $t>0$ by Logarithmic Sobolev inequalities for $H=-Δ+ q(x)$ 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 $\mathbb{R}^{n}$.

Theorems & Definitions (14)

• Lemma 1.1
• Lemma 2.1
• Remark 2.2
• proof : Proof of Lemma \ref{['Rosen_inequalities']}
• Lemma 2.3: Rosen's Lemma
• Theorem 3.1
• Corollary 3.2
• Remark 4.1
• Definition A.1
• Theorem A.2: Agmon-type comparison principle