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 lenghth L in R. We show that if Γ is C-smooth and has regular endpoints, the j-th eigenvalue of such an operator has the asymptotic expansion λj(Hα,Γ) = ξα+λj(S)+O(e) as the coupling parameter α → ∞, where ξα = −4 e2(−2πα+ψ(1)) and λj(S) is the j-th eigenvalue of the Schrödinger operator S = − d 2 ds2 − 1 4 γ(s) on L(0, L) with Dirichlet condition at the interval endpoints in which γ is the curvature of Γ. Mathematics Subject Classification (2010): Primary 81Q10; Secondary 35J10, 35P15