On absence of bound states for weakly attractive $\delta^\prime$-interactions supported on non-closed curves in $\mathbb{R}^2$

Abstract

Let $\Lambda\subset\mathbb{R}^2$ be a non-closed piecewise-$C^1$ curve, which is either bounded with two free endpoints or unbounded with one free endpoint. Let $u_\pm|_\Lambda \in L^2(\Lambda)$ be the traces of a function $u$ in the Sobolev space $H^1({\mathbb R}^2\setminus \Lambda)$ onto two faces of $\Lambda$. We prove that for a wide class of shapes of $\Lambda$ the Schr\"odinger operator $\mathsf{H}_\omega^\Lambda$ with $\delta^\prime$-interaction supported on $\Lambda$ of strength $\omega \in L^\infty(\Lambda;\mathbb{R})$ associated with the quadratic form \[ H^1(\mathbb{R}^2\setminus\Lambda)\ni u \mapsto \int_{\mathbb{R}^2}\big|\nabla u \big|^2 \mathsf{d} x - \int_\Lambda \omega \big| u_+|_\Lambda - u_-|_\Lambda \big|^2 \mathsf{d} s \] has no negative spectrum provided that $\omega$ is pointwise majorized by a strictly positive function explicitly expressed in terms of $\Lambda$. If, additionally, the domain $\mathbb{R}^2\setminus\Lambda$ is quasi-conical, we show that $\sigma(\mathsf{H}_\omega^\Lambda) = [0,+\infty)$. For a bounded curve $\Lambda$ in our class and non-varying interaction strength $\omega\in\mathbb{R}$ we derive existence of a constant $\omega_* > 0$ such that $\sigma(\mathsf{H}_\omega^\Lambda) = [0,+\infty)$ for all $\omega \in (-\infty, \omega_*]$; informally speaking, bound states are absent in the weak coupling regime.