On absence of bound states for weakly attractive δ′-interactions supported on non-closed curves in ℝ2

Abstract

Let Λ ⊂ ℝ2 be a non-closed piecewise-C1 curve, which is either bounded with two free endpoints or unbounded with one free endpoint. Let u±|Λ ∈ L2(Λ) be the traces of a function u in the Sobolev space H1(ℝ2∖Λ) onto two faces of Λ. We prove that for a wide class of shapes of Λ the Schrödinger operator HωΛ with δ′-interaction supported on Λ of strength ω ∈ L∞(Λ; ℝ) associated with the quadratic form H1(R2∖Λ)∋u↦∫R2∇u2dx−∫Λωu+|Λ−u−|Λ2ds 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 ℝ2∖Λ is quasi-conical, we show that σ(HωΛ)=[0,+∞). For a bounded curve Λ in our class and non-varying interaction strength ω ∈ ℝ, we derive existence of a constant ω∗ > 0 such that σ(HωΛ)=[0,+∞) for all ω ∈ (−∞, ω∗]; informally speaking, bound states are absent in the weak coupling regime.