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

TL;DR

This paper establishes conditions on the potential in Schr"odinger operators that lead to Logarithmic Sobolev inequalities, enabling the proof of intrinsic ultracontractivity of the associated semigroup in $L^2(\mathbb{R}^n)$.

Contribution

It introduces growth conditions on the potential ensuring Rosen inequalities and Logarithmic Sobolev inequalities, which are used to prove intrinsic ultracontractivity of Schr"odinger semigroups.

Findings

Rosen inequalities for the ground state under specific growth conditions

Logarithmic Sobolev inequalities derived from Rosen inequalities

Proof of intrinsic ultracontractivity of Schr"odinger semigroups

Abstract

In the first part of this article we present a growth condition on the potential $q$ in the Schr\"odinger operator $H=-\Delta + q(x)$ in $\mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ that implies Rosen inequalities for the ground state $\varphi$ of $H$, i.e. $\forall \varepsilon > 0 \exists \gamma(\varepsilon) > 0 \ : \ - \ln\left( \varphi(x) \right) \leq \varepsilon q(x) + \gamma(\varepsilon)$. 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 Schr\"odinger 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 Schr\"odinger 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 Schr\"odinger semigroups and Logarithmic Sobolev inequalities.