关于Schrödinger半群在L²(Rⁿ)中内在超收缩性的内在超收缩性：通过对数Sobolev不等式
Intrinsic Ultracontractivity for a class of Schroedinger Semigroups in $L^{2}(\mathbb{R}^{n})$ by Logarithmic Sobolev inequalities
AI总结：
本文通过研究Schrödinger算子的基态和对数Sobolev不等式，证明了在L²(Rⁿ)中Schrödinger半群的内在超收缩性。
摘要：
在本文的第一部分中，我们给出了Schrödinger算子H=-Δ+q(x)在L²(Rⁿ)中势函数q的增长条件，该条件暗示了对于H的基态φ的Rosen不等式，即对于任意ε>0存在γ(ε)>0使得- ln(φ(x)) ≤ εq(x)+γ(ε)。虽然这些不等式本身并不特别有趣，但它们提供了绝对必要的对数Sobolev不等式，用于证明所关联的Schrödinger半群e^{-tH}的内在超收缩性，即对于任意t>0存在C_t>0使得对于每一个u∈L²(Rⁿ)几乎处处在Rⁿ中成立|e^{-tH}u(x)| ≤ C_tφ(x)||u||₂。在本文的第二部分中，我们证明了这一点。为了证明Rosen不等式，我们专注于解决一个径向Schrödinger不等式，并使用Agmon的比较原理版本和Young关于递增函数的不等式。我们遵循经典的证明内在超收缩性e^{-tH}的方法，使用加权Sobolev函数空间、加权Schrödinger半群和对数Sobolev不等式。
英文摘要：
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 $φ$ of $H$, i.e. $\forall \varepsilon > 0 \exists γ(\varepsilon) > 0 \ : \ - \ln\left( φ(x) \right) \leq \varepsilon q(x) + γ(\varepsilon)$. 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. $\forall t>0 \exists C_{t} > 0 \ : \ \left| \mathrm{e}^{-tH} u (x) \right| \ \leq \ C_{t} φ(x) \| u \|_{2}$ 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.
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