Strong Coupling Asymptotics for Schrödinger Operators with an Interaction Supported by an Open Arc in three Dimensions

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• 作者：Pavel Exner ^{exner@ujf.cas.cz" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Primary 81Q10 ; Secondary 35J10 ; 35P15
• 刊名：Reports on Mathematical Physics
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：77
• 期：1
• 页码：1-17
• 全文大小：156 K

文摘

We consider Schrödinger operators with a strongly attractive singular interaction supported by a finite curve Γ of length L in ℝ³. We show that if Γ is C⁴-smooth and has regular endpoints, the j  -th eigenvalue of such an operator has the asymptotic expansionλ_j(H_α,Γ)=ξ_α+λ_j(S)+O(eπα)${\lambda }_j\left(H_{\alpha ,\Gamma }\right)={\xi }_{\alpha }+{\lambda }_j\left(S\right)+\mathcal{O}\left({\text{e}}^{\pi \alpha }\right)$ as the coupling parameter α → -∞, where ξ_α = −4e^{2(−2πα+ψ(1))} and λ_j(S) is the j  -th eigenvalue of the Schrödinger operator$S=-\frac{d^2}{d{s}^2}-\frac{1}{4}{\gamma }^2\left(s\right)$on L²(0, L) with Dirichlet condition at the interval endpoints in which γ is the curvature of Γ.