Strong-coupling asymptotic expansion for Schrödinger operators with a singular interaction supported by a curve in $\mathbb{R}^3$

Abstract

We investigate a class of generalized Schrödinger operators in $L^2(\mathbb{R}^3)$ with a singular interaction supported by a smooth curve $Γ$. We find a strong-coupling asymptotic expansion of the discrete spectrum in case when $Γ$ is a loop or an infinite bent curve which is asymptotically straight. It is given in terms of an auxiliary one-dimensional Schrödinger operator with a potential determined by the curvature of $Γ$. In the same way we obtain an asymptotics of spectral bands for a periodic curve. In particular, the spectrum is shown to have open gaps in this case if $Γ$ is not a straight line and the singular interaction is strong enough.