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 $H=-\Delta +q(x)$ in ${\mathrm{L}}^2\left({\mathbb{R}}^n\right)$ that implies Rosen inequalities for the ground state $\phi$ of $H$, i.e. $\forall \epsilon >0\exists \gamma (\epsilon )>0:-\ln \left(\phi (x)\right)\le \epsilon q(x)+\gamma (\epsilon )$. 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. $\forall t>0\exists C_t>0:\left|{\mathrm{e}}^{-tH}u(x)\right|\le C_t\phi (x)\Vert u{\Vert }_2$ 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.