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 $q:{\mathbb{R}}^n\to (0,\infty )$ that implies the weighted Schrödinger semigroup ${\phi }^{-1}{\mathrm{e}}^{-tH}\phi$ to map a weighted Lebesgue function space ${\mathrm{L}}_{\mu }^1({\mathbb{R}}^n)$ into a weighted Lebesgue function space ${\mathrm{L}}_{\mu }^2({\mathbb{R}}^n)$ continously at every time $t>0$ by Logarithmic Sobolev inequalities for $H=-\Delta +q(x)$ with it's strictly positive ground state $\phi :{\mathbb{R}}^n\to (0,\infty )$. We use the self-adjointness of ${\mathrm{e}}^{-tH}$ in ${\mathrm{L}}^2({\mathbb{R}}^n)$ to infer an intrinsic ultracontractivity, i.e. $\forall t>0\exists C_t>0:\left|{\mathrm{e}}^{-tH}u(x)\right|\le C_t\phi (x)\Vert u{\Vert }_2$ for every $u\in {\mathrm{L}}^2\left({\mathbb{R}}^n\right)$ almost everywhere in ${\mathbb{R}}^n$.