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

Let $\mathrm{\Lambda }\subset {\mathbb{R}}^2$ be a non-closed piecewise-$C^1$ curve, which is either bounded with two free endpoints or unbounded with one free endpoint. Let $u_{\pm }{\mathrm{\mid }}_{\mathrm{\Lambda }}\in L^2(\mathrm{\Lambda })$ be the traces of a function $u$ in the Sobolev space $H^1({\mathbb{R}}^2\setminus \mathrm{\Lambda })$ onto two faces of $\mathrm{\Lambda }$. We prove that for a wide class of shapes of $\mathrm{\Lambda }$ the Schr"odinger operator ${\mathsf{H}}_{\omega }^{\mathrm{\Lambda }}$ with ${\delta }^{\mathrm{\prime }}$-interaction supported on $\mathrm{\Lambda }$ 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 $\mathrm{\Lambda }$. If, additionally, the domain ${\mathbb{R}}^2\setminus \mathrm{\Lambda }$ is quasi-conical, we show that $\sigma ({\mathsf{H}}_{\omega }^{\mathrm{\Lambda }})=[0,+\mathrm{\infty })$. For a bounded curve $\mathrm{\Lambda }$ in our class and non-varying interaction strength $\omega \in \mathbb{R}$ we derive existence of a constant ${\omega }_{\ast }>0$ such that $\sigma ({\mathsf{H}}_{\omega }^{\mathrm{\Lambda }})=[0,+\mathrm{\infty })$ for all $\omega \in (-\infty, \omega*]$; informally speaking, bound states are absent in the weak coupling regime.