$\mathbb{R}^2$中非闭曲线支持的弱吸引$δ^\prime$ -相互作用的束缚态缺失
On absence of bound states for weakly attractive $δ^\prime$-interactions supported on non-closed curves in $\mathbb{R}^2$
摘要：
设$Λ\subset\mathbb{R}^2$是一条非封闭的分段$C^1$曲线，它要么有两个自由端点有界，要么有一个自由端点无界。令$u_\pm|_Λ\in L^2(Λ)$是函数$u$在Sobolev空间$H^1({\mathbb R}^2\setminus Λ)$中到$Λ$的两个面上的迹。我们证明了，对于广泛的一类形状的$Λ$的薛定谔算子$\mathsf{H}_ω^Λ$与$δ^\prime$ -相互作用支持$Λ$的强度$ω\in L^\infty(Λ;\mathbb{R})$与二次型\[ H^1(\mathbb{R}^2\setminusΛ)\ni u \mapsto \int_{\mathbb{R}^2}\big|\nabla u \big|^2 \mathsf{d} x - \int_Λω\big| u_+|_Λ- u_-|_Λ\big|^2 \mathsf{d} s \]没有负谱，只要$ω$是逐点优化的严格正的功能明确表示的$Λ$。如果，此外，域$\mathbb{R}^2\setminusΛ$是准圆锥形，我们表明，$σ(\mathsf{H}_ω^Λ) = [0,+\infty)$。对于有界曲线$Λ$在我们的类和不变的相互作用强度$ω\in\mathbb{R}$，我们推导出一个常数$ω_* > 0$的存在，使$σ(\mathsf{H}_ω^Λ) = [0,+\infty)$的所有$ω\in (-\infty, ω_*]$ ;非正式地说，束缚态是在弱耦合制度缺席。
英文摘要：
Let $Λ\subset\mathbb{R}^2$ be a non-closed piecewise-$C^1$ curve, which is either bounded with two free endpoints or unbounded with one free endpoint. Let $u_\pm|_Λ\in L^2(Λ)$ be the traces of a function $u$ in the Sobolev space $H^1({\mathbb R}^2\setminus Λ)$ onto two faces of $Λ$. We prove that for a wide class of shapes of $Λ$ the Schrödinger operator $\mathsf{H}_ω^Λ$ with $δ^\prime$-interaction supported on $Λ$ of strength $ω\in L^\infty(Λ;\mathbb{R})$ associated with the quadratic form \[ H^1(\mathbb{R}^2\setminusΛ)\ni u \mapsto \int_{\mathbb{R}^2}\big|\nabla u \big|^2 \mathsf{d} x - \int_Λω\big| u_+|_Λ- u_-|_Λ\big|^2 \mathsf{d} s \] has no negative spectrum provided that $ω$ is pointwise majorized by a strictly positive function explicitly expressed in terms of $Λ$. If, additionally, the domain $\mathbb{R}^2\setminusΛ$ is quasi-conical, we show that $σ(\mathsf{H}_ω^Λ) = [0,+\infty)$. For a bounded curve $Λ$ in our class and non-varying interaction strength $ω\in\mathbb{R}$ we derive existence of a constant $ω_* > 0$ such that $σ(\mathsf{H}_ω^Λ) = [0,+\infty)$ for all $ω\in (-\infty, ω_*]$; informally speaking, bound states are absent in the weak coupling regime.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
