On absence of bound states for weakly attractive \(\delta^{\prime}\)-interactions supported on non-closed curves in \(\mathbb{R}^{2}\) (Q2795538)

For a non-closed piecewise-\(C^1\) curve \(\Lambda \in \mathbb{R}^2\) and \(\omega \in L^{\infty}(\Lambda ; \mathbb{R})\), the sesquilinear form \(\alpha^{\Lambda}_{\omega}\) is defined as NEWLINENEWLINE\[NEWLINE\alpha^{\Lambda}_{\omega}[u,v]:=(\nabla u, \nabla v)_{\mathbb{R}^2}-(\omega[u]_{\Lambda}, [v]_{\Lambda})_{\Lambda},NEWLINE\]NEWLINE NEWLINEwhere \([u]_{\Lambda}:=u_+|_{\Lambda}-u_-|_{\Lambda}\). This paper deals with the Schrödinger operator \(H^{\Lambda}_{\omega}\) with \({\delta}^{\prime}\)-interaction supported on a non-closed \(C^1\)-curve \(\Lambda\) of strength \(\omega \in L^{\infty}(\Lambda ; \mathbb{R})\) associated with the sesquilinear form \(\alpha^{\Lambda}_{\omega}\). Particularly, the authors analyze the types of \(\Lambda\) for which \(H^{\Lambda}_{\omega}\) has no negative spectrum and thus it has no bound states. Their results claim that if \(\mathbb{R}/\Lambda\) is quasi-conical, \(\sigma(H^{\Lambda}_{\omega})=[0,\infty)\). For a bounded curve \(\Lambda\) and a constant \(\omega \in \mathbb{R}\), there is a \(\omega_*>0\) such that NEWLINENEWLINE\[NEWLINE\sigma(H^{\Lambda}_{\omega})=[0,\infty)\;\;\text{ for all}\;\;\omega\in (-\infty, \omega_*].NEWLINE\]NEWLINE NEWLINEThe paper concludes with posing open questions. One of them is to find the critical strength \(\omega_*\) for other types of \(\Lambda\) such that \(\sigma(H^{\Lambda}_{\omega})=[0,\infty)\) for all \(\omega\in (-\infty, \omega_*]\).