Intrinsic Ultracontractivity for a class of Schroedinger Semigroups in $L^{2}(\mathbb{R}^{n})$ by Logarithmic Sobolev inequalities

Abstract: In the first part of this article we present a growth condition on the potential $q$ in the Schrödinger operator $H=-Δ+ q(x)$ in $\mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ that implies Rosen inequalities for the ground state $\varphi$ of $H$, i.e. $\forall \varepsilon > 0 \exists γ(\varepsilon) > 0 \ : \ - \ln\left( \varphi(x) \right) \leq \varepsilon q(x) + γ(\varepsilon)$. While these inequalities are not particularly interesting in themselves, they offer Logarithmic Sobolev inequalities which are absolutely essential to prove an intrinsic ultracontractivity of the associated Schrödinger semigroup $\mathrm{e}^{-tH}$, i.e. $\forall t>0 \exists C_{t} > 0 \ : \ \left| \mathrm{e}^{-tH} u (x) \right| \ \leq \ C_{t} \varphi(x) | u |_{2}$ holds for every $u \in \mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ almost everywhere in $\mathbb{R}^{n}$ which we prove in the second part of this article. For proving Rosen inequalities we focus on solving a radial Schrödinger inequality and use Agmon's version of the comparison principle and Young's inequality for increasing functions. We follow the classic method proving intrinsic ultracontractivity of $\mathrm{e}^{-tH}$ by using weighted Sobolev function spaces, weighted Schrödinger semigroups and Logarithmic Sobolev inequalities.