无界域上带点相互作用的Robin拉普拉斯算子：离散谱与耦合渐近行为
Robin Laplacians with point interactions on unbounded domains: discrete spectrum and coupling asymptotics
- University of Milano–Bicocca（米兰比可卡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究二维、三维无界域及外域上带有限多点相互作用的Neumann、Robin拉普拉斯算子的离散谱，推导本征值计数公式与渐近行为，通过外球、外盘模型阐明临界耦合特性。
AI中文摘要：
我们在二维和三维的特殊无界域及外域C^{1,1}域中，研究Neumann和Robin拉普拉斯算子的有限多个点相互作用的离散谱。这些算子通过普通边界三元组实现为自伴延拓，其伽马场和Weyl矩阵由Robin格林核构造。背景谱以下的本征值由Weyl矩阵表征，得到精确的有限维计数公式。若背景算子是非负的，这也能给出负本征值的数量，无需假设Weyl矩阵存在有限的零能极限。在单中心情形下，我们确定了临界耦合，并证明唯一的本征值分支是实解析的、严格递增且严格凹的。对于标量多中心耦合Θ=αI_N，足够强的吸引作用会在背景谱以下产生恰好N个本征值，且所有这些本征值具有与全空间规律一致的普适主导渐近行为。若背景谱的底部是孤立本征值，则该分支对每个有限耦合都存在，我们确定了当耦合趋于+∞时其主导的退耦渐近行为。明确的外球和外盘模型阐明了临界耦合及其阈值行为。
英文摘要：
We investigate the discrete spectrum of finitely many point interactions for Neumann and Robin Laplacians on special unbounded and exterior $C^{1,1}$ domains in dimensions two and three. The operators are realized as self-adjoint extensions through an ordinary boundary triple whose gamma field and Weyl matrix are constructed from the Robin Green kernel. Eigenvalues below the background spectrum are characterized by the Weyl matrix, yielding an exact finite-dimensional counting formula. If the background operator is non-negative, this also gives the number of negative eigenvalues without assuming a finite zero-energy limit of the Weyl matrix. In the one-centre case we identify the critical coupling and prove that the unique eigenvalue branch is real analytic, strictly increasing, and strictly concave. For scalar multicentre couplings $Θ=αI_N$, sufficiently strong attraction produces exactly $N$ eigenvalues below the background spectrum, and all of them have universal leading asymptotics coinciding with the whole-space laws. If the bottom of the background spectrum is an isolated eigenvalue, the branch exists for every finite coupling and we determine its leading decoupling asymptotics as the coupling tends to $+\infty$. Explicit exterior-sphere and exterior-disk models illustrate the critical couplings and their threshold behaviour.
