Abstract Let Γ ⊂ 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 Schrodinger operator H α with the attractive δ-potential of strength α > 0 supported by Γ, which is defined by its quadratic form H 1 ( R 2 ) ∋ u ↦ ∬ R 2 | ∇ u | 2 d x − α ∫ Γ u 2 d s , where d s stands for the one-dimensional Hausdorff measure on Γ. It is shown that if n ∈ N is fixed and α is large, then the well-defined nth eigenvalue E n ( H α ) of H α behaves as E n ( H α ) = − α 2 + 2 2 p + 2 E n α 6 p + 2 + O ( α 6 p + 2 − η ) , where the constants E n > 0 are the eigenvalues of an explicitly given one-dimensional Schrodinger 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.

中文翻译：

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由带尖点的曲线支持的 δ 相互作用的强耦合渐近性

摘要 设Γ ⊂ R 2 是一条简单的闭合曲线，除了原点外是光滑的，它有一个幂尖点并与曲线|重合。× 2 | = x 1 p 对于某些 p > 1 。我们研究了薛定谔算子 H α 的特征值，其强度 α > 0 的吸引力 δ 势由 Γ 支持，Γ 由其二次形式 H 1 ( R 2 ) ∋ u ↦ ∬ R 2 | 定义。∇ 你 | 2 dx − α ∫ Γ u 2 ds ，其中ds 代表Γ 上的一维Hausdorff 测度。结果表明，如果 n ∈ N 固定且 α 较大，则 H α 的明确定义的第 n 个特征值 E n ( H α ) 表现为 E n ( H α ) = − α 2 + 2 2 p + 2 E n α 6 p + 2 + O ( α 6 p + 2 − η ) ，其中常数 E n > 0 是由尖点确定的明确给定的一维薛定谔算子的特征值，并且 η > 0 。