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 Schrödinger 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 \[ H^1(\mathbb{R}^2\setminusΛ)\ni u \mapsto \int_{\mathbb{R}^2}\big|\nabla u \big|^2 \mathsf{d} x - \int_Λω\big| u_+|_Λ- u_-|_Λ\big|^2 \mathsf{d} s \] has no negative spectrum provided that $\omega$ 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 ${\omega }_{\ast }>0$ such that $\sigma ({\mathsf{H}}_{\omega }^{\Lambda })=[0,+\infty )$ for all $\omega \in (-\infty ,{\omega }_{\ast }]$; informally speaking, bound states are absent in the weak coupling regime.

22 pages, 2 figures

References in corpus (2)

• Schrödinger operators with delta and delta'-potentials supported on hypersurfaces
• Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces

Cited by in corpus (3)

• Generalized interactions supported on hypersurfaces
• 2D Schrödinger operators with singular potentials concentrated near curves
• Trace Hardy inequality for the Euclidean space with a cut and its applications