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

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 expansion λj(Hα,Γ)=ξα+λj(S)+O(eπα)λj(Hα,Γ)=ξα+λj(S)+O(eπα) as the coupling parameter α → -∞, where ξα = −4e2(−2πα+ψ(1)) and λj(S) is the j  -th eigenvalue of the Schrödinger operator S=−d2ds2−14γ2(s)on L2(0, L) with Dirichlet condition at the interval endpoints in which γ is the curvature of Γ.