Intrinsic Ultracontractivity for a class of Schroedinger Semigroups in $\mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ using Log-Sobolev-inequalities and duality arguments
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| Format: | Preprint |
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2026
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| author | Schwerdt, Christoph Ouelddris, Ilham |
| author_facet | Schwerdt, Christoph Ouelddris, Ilham |
| contents | We present a class of potentials $q \colon \mathbb{R}^{n} \to (0,\infty)$ that implies the weighted Schrödinger semigroup $φ^{-1}\mathrm{e}^{-tH}φ$ to map a weighted Lebesgue function space $\mathrm{L}_μ^{1}(\mathbb{R}^{n})$ into a weighted Lebesgue function space $\mathrm{L}_μ^{2}(\mathbb{R}^{n})$ continously at every time $t>0$ by Logarithmic Sobolev inequalities for $H=-Δ+ q(x)$ with it's strictly positive ground state $φ\colon \mathbb{R}^{n} \to (0,\infty)$. We use the self-adjointness of $\mathrm{e}^{-tH}$ in $\mathrm{L}^{2}(\mathbb{R}^{n})$ to infer an intrinsic ultracontractivity, i.e.
$\forall t>0 \ \exists C_{t} > 0 \ : \ \left| \mathrm{e}^{-tH} u (x) \right| \ \leq \ C_{t} φ(x) \| u \|_{2}$
for every $u \in \mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ almost everywhere in $\mathbb{R}^{n}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_04738 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Intrinsic Ultracontractivity for a class of Schroedinger Semigroups in $\mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ using Log-Sobolev-inequalities and duality arguments Schwerdt, Christoph Ouelddris, Ilham Analysis of PDEs Mathematical Physics Functional Analysis We present a class of potentials $q \colon \mathbb{R}^{n} \to (0,\infty)$ that implies the weighted Schrödinger semigroup $φ^{-1}\mathrm{e}^{-tH}φ$ to map a weighted Lebesgue function space $\mathrm{L}_μ^{1}(\mathbb{R}^{n})$ into a weighted Lebesgue function space $\mathrm{L}_μ^{2}(\mathbb{R}^{n})$ continously at every time $t>0$ by Logarithmic Sobolev inequalities for $H=-Δ+ q(x)$ with it's strictly positive ground state $φ\colon \mathbb{R}^{n} \to (0,\infty)$. We use the self-adjointness of $\mathrm{e}^{-tH}$ in $\mathrm{L}^{2}(\mathbb{R}^{n})$ to infer an intrinsic ultracontractivity, i.e. $\forall t>0 \ \exists C_{t} > 0 \ : \ \left| \mathrm{e}^{-tH} u (x) \right| \ \leq \ C_{t} φ(x) \| u \|_{2}$ for every $u \in \mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ almost everywhere in $\mathbb{R}^{n}$. |
| title | Intrinsic Ultracontractivity for a class of Schroedinger Semigroups in $\mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ using Log-Sobolev-inequalities and duality arguments |
| topic | Analysis of PDEs Mathematical Physics Functional Analysis |
| url | https://arxiv.org/abs/2602.04738 |