Schrödinger算子在具有缩放共振势域上的收敛性
Convergence of Schrödinger operators on domains with scaled resonant potentials
AI总结：
研究Schrödinger算子在具有缩放共振势的域上的收敛性，揭示共振和非共振情况下算子极限的不同表现及可能的假设省略
摘要：
我们考虑在具有Dirichlet边界条件的有界光滑域（维度d≥2）上，具有适当缩放势的Schrödinger算子。该势仅依赖于域边界的距离。我们的目标是分析当缩放参数趋于零时这些算子的收敛性。如果缩放势是共振的，强余弦意义下的极限是一个Robin拉普拉斯算子，其边界系数由边界曲率的均值表示。一个反例表明，在这种情况下，规范余弦收敛一般不成立。如果缩放势是非共振的，并且满足关于负部分小性的显式假设，强余弦意义下的极限是Dirichlet拉普拉斯算子。我们猜想在非共振情况下可以省略这个额外的假设。
英文摘要：
We consider Schrödinger operators on a bounded, smooth domain of dimension $d \ge 2$ with Dirichlet boundary conditions and a properly scaled potential, which depends only on the distance to the boundary of the domain. Our aim is to analyse the convergence of these operators as the scaling parameter tends to zero. If the scaled potential is resonant, the limit in strong resolvent sense is a Robin Laplacian with boundary coefficient expressed in terms of the mean curvature of the boundary. A counterexample shows that norm resolvent convergence cannot hold in general in this setting. If the scaled potential is non-resonant and satisfies an explicit assumption on the smallness of the negative part, the limit in strong resolvent sense is the Dirichlet Laplacian. We conjecture that we can drop this additional assumption in the non-resonant case.
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