Intrinsic Ultracontractivity for a class of Schroedinger Semigroups in ${\mathrm{L}}^2\left({\mathbb{R}}^n\right)$ using Log-Sobolev-inequalities and duality arguments

We present a class of potentials $q\text{\hspace{0.17em}⁣}:{\mathbb{R}}^n\to (0,\mathrm{\infty })$ 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 $H=-\mathrm{\Delta }+q(x)$ with it’s strictly positive ground state $\phi \text{\hspace{0.17em}⁣}:{\mathbb{R}}^n\to (0,\mathrm{\infty })$. We use the self-adjointness of ${\mathrm{e}}^{-tH}$ in ${\mathrm{L}}^2({\mathbb{R}}^n)$ to infer an intrinsic ultracontractivity, i.e. $\mathrm{\forall }t>0\mathrm{\exists }{C}_t>0:\mid {\mathrm{e}}^{-tH}u(x)\mid \le C_t\phi (x)\mathrm{\parallel }u{\mathrm{\parallel }}_2$ for every $u\in {\mathrm{L}}^2\left({\mathbb{R}}^n\right)$ almost everywhere in ${\mathbb{R}}^n$.