量子图顶点耦合的奇异标度势近似
Approximations of quantum-graph vertex couplings by singularly scaled potentials
摘要：
研究了一类$H_\varepsilon= -\frac{\mathrm{d}^2}{\mathrm{d}x^2}+ \frac{λ(\varepsilon)}{\varepsilon^2}Q \big(\frac{x}{\varepsilon}\big)$形式的薛定谔算子作用在顶点满足Kirchhoff条件的$n$ -边星图上的极限性质.实值位势$Q$被假定为具有紧支集，$λ(\cdot)$被假定为在$\varepsilon=0$周围解析的，且具有$λ(0)=1$ .我们表明，如果运营商有一个零能量共振$m$顺序$\varepsilon=1$和$λ(1)=1$，在极限$\varepsilon\to 0$得到的拉普拉斯算子与顶点耦合取决于$1+\frac12 m(2n-m+1)$参数。我们证明了范数预解式的收敛性以及相应的壳散射矩阵的收敛性。所得到的顶点耦合是尺度不变的类型提供$λ'(0)=0$，否则散射矩阵依赖于能量和标度势变得渐近不透明的低能量极限。
英文摘要：
We investigate the limit properties of a family of Schrödinger operators of the form $H_\varepsilon= -\frac{\mathrm{d}^2}{\mathrm{d}x^2}+ \frac{λ(\varepsilon)}{\varepsilon^2}Q \big(\frac{x}{\varepsilon}\big)$ acting on $n$-edge star graphs with Kirchhoff conditions imposed at the vertex. The real-valued potential $Q$ is supposed to have compact support and $λ(\cdot)$ to be analytic around $\varepsilon=0$ with $λ(0)=1$. We show that if the operator has a zero-energy resonance of order $m$ for $\varepsilon=1$ and $λ(1)=1$, in the limit $\varepsilon\to 0$ one obtains the Laplacian with a vertex coupling depending on $1+\frac12 m(2n-m+1)$ parameters. We prove the norm-resolvent convergence as well as the convergence of the corresponding on-shell scattering matrices. The obtained vertex couplings are of scale-invariant type provided $λ'(0)=0$; otherwise the scattering matrix depends on energy and the scaled potential becomes asymptotically opaque in the low-energy limit.
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。
