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

TL;DR

This paper proves that for certain non-closed curves in , the Schrf6dinger operator with  ext{-}interaction supported on the curve has no negative spectrum if the interaction strength is sufficiently positive, indicating absence of bound states in weak coupling.

Contribution

It establishes conditions under which  ext{-}interactions on non-closed curves do not produce bound states, extending understanding of spectral properties of such operators.

Findings

No negative spectrum for sufficiently positive interaction strength.

Spectral equality (}^2b7) for quasi-conical domains.

Existence of a threshold b7_* for weak coupling regimes.

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.