尖点曲线支持的$δ$ -相互作用的强耦合渐近性
Strong coupling asymptotics for $δ$-interactions supported by curves with cusps
摘要：
设$Γ\subset \mathbb{R}^2$是一条简单的闭曲线，除了在原点处是光滑的，在原点处它有一个幂尖点，并且对于某些$p>1$，它与曲线$|x_2|=x_1^p$重合。研究了具有$Γ$支持的吸引$δ$ -强度势$α>0$的Schr dinger算子$H_α$的特征值，$α>0$由其二次形式\[ H^1(\mathbb{R}^2)\ni u\mapsto \iint_{\mathbb{R}^2} |\nabla u|^2\,\mathrm{d}x-α\int_Γu^2\, \mathrm{d}s, \]定义，其中$\mathrm{d}s$表示$Γ$上的一维Hausdorff测度.证明了如果$n\in\mathbb{N}$是固定的，$α$是大的，则$H_α$的定义良好的$n$第th特征值$E_n(H_α)$表现为\[ E_n(H_α)=-α^2 + 2^{\frac{2}{p+2}} \mathcal{E}_n \,α^{\frac{6}{p+2}} + \mathcal{O}(α^{\frac{6}{p+2}-η}), \]，其中常数$\mathcal{E}_n>0$是由尖点和$η>0$确定的显式给定的一维薛定谔算子的特征值.主项和次项在这个渐近展开是不同于以前所观察到的情况下，$Γ$是光滑的或分段光滑的非零角度。
英文摘要：
Let $Γ\subset \mathbb{R}^2$ be a simple closed curve which is smooth except at the origin, at which it has a power cusp and coincides with the curve $|x_2|=x_1^p$ for some $p>1$. We study the eigenvalues of the Schrödinger operator $H_α$ with the attractive $δ$-potential of strength $α>0$ supported by $Γ$, which is defined by its quadratic form \[ H^1(\mathbb{R}^2)\ni u\mapsto \iint_{\mathbb{R}^2} |\nabla u|^2\,\mathrm{d}x-α\int_Γu^2\, \mathrm{d}s, \] where $\mathrm{d}s$ stands for the one-dimensional Hausdorff measure on $Γ$. It is shown that if $n\in\mathbb{N}$ is fixed and $α$ is large, then the well-defined $n$th eigenvalue $E_n(H_α)$ of $H_α$ behaves as \[ E_n(H_α)=-α^2 + 2^{\frac{2}{p+2}} \mathcal{E}_n \,α^{\frac{6}{p+2}} + \mathcal{O}(α^{\frac{6}{p+2}-η}), \] where the constants $\mathcal{E}_n>0$ are the eigenvalues of an explicitly given one-dimensional Schrödinger operator determined by the cusp, and $η>0$. Both main and secondary terms in this asymptotic expansion are different from what was observed previously for the cases when~$Γ$ is smooth or piecewise smooth with non-zero angles.
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。
