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

Abstract

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