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

TL;DR

This work derives intrinsic ultracontractivity for Schrödinger semigroups $e^{-tH}$ on $L^2(\mathbb{R}^n)$ by establishing Rosen inequalities for the ground state under explicit growth bounds on the potential $q(x)$. The authors leverage a radial Schrödinger inequality, Agmon's comparison principle, and Young's inequality for increasing functions to obtain Rosen bounds $-\ln(\varphi(x))\le\varepsilon q(x)+\gamma(\varepsilon)$ via a controllable upper comparison $Q$ and auxiliary functions $f_{k,m-1}$. These Rosen inequalities feed into weighted Logarithmic Sobolev inequalities for the associated weighted semigroup, enabling a bootstrap to intrinsic ultracontractivity: $|e^{-tH}u(x)|\le C_t\varphi(x)\|u\|_2$, with the ground state $\varphi$ dominating the asymptotic behavior. The paper also provides concrete classes of bounding potentials $Q$ (and related $q$) that satisfy the hypotheses, offering a practical framework for verifying IU in nonradial settings. The approach integrates weighted function spaces, comparison principles, and detailed radial analysis to yield explicit, verifiable conditions for IU in Schrödinger semigroups.

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.

Theorems & Definitions (26)

• Lemma 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Remark 2.3
• Theorem 3.1
• Remark 3.2
• Lemma 3.3
• Lemma 5.1