Curvature-Induced Bound States for a $ \delta $ Interaction Supported by a Curve in $ \mathbb{R}^3 $

Abstract

We study the Laplacian in L 2 (Ê 3 ) perturbed on an infinite curve Γ by a δ interaction defined through boundary conditions which relate the corresponding generalized boundary values. We show that if Γ is smooth and not a straight line but it is asymptotically straight in a suitable sense, and if the interaction does not vary along the curve, the perturbed operator has at least one isolated eigenvalue below the threshold of the essential spectrum. P. Exner and S. KondejAnn. Henri Poincaré other tools. One possibility is to employ the resolvent formula for a curve in R 3 derived in [Ku]. However, since it uses rather strong regularity hypotheses about the curve we take another route and begin instead with an abstract formula for strongly singular perturbations due to A. Posilicano [Po1]. When it is specified to our particular case, it contains again an embedding operator into a space of functions supported on the curve Γ, however, this time it is not the "naive" L 2 but rather a suitable element from the scale of Sobolev spaces. Of course, one can regard it as a generalization of Krein's formula; recall that such a way of expressing the resolvent can be used not only to describe δ interaction perturbations but also more general dynamics supported by zero measure sets [Ka, KK, Ko]. Another aspect of the absence of a description in terms of the quadratic-form sum concerns the very definition of the operator we want to study. We have to employ boundary conditions which relate the corresponding generalized boundary values in the normal plane to the curve modeled after the usual two-dimensional δ interaction [AGHH], which requires us to impose stronger regularity conditions on Γ. Furthermore, a modification of the Birman-Schwinger technique used in [EI] demands stronger restrictions on the regularity of the curve. On the other hand, apart of these technical hypotheses our main result -stated in Theorem 5.6 below -is analogous to that of [EI], namely that for any curve which is asymptotically straight but not a straight line the corresponding operator has at least one isolated eigenvalue. This conclusion is by no means obvious having in mind how different are the point interactions in one and two dimensions.which are obviously bounded too. Let Z be an open subset of (−∆) symmetric w.r.t. the real axis, i.e. such that z ∈ Z impliesz ∈ Z. Suppose that for any Vol. 3, 2002 Curvature-Induced Bound States 969 z ∈ Z there exists a closed operator Q z : D ⊆ X → X satisfying the following conditions,

Absolute Continuity of the Spectrum for Periodically Modulated Leaky Wires in $${\mathbb{R}^{3}}$$

“…Moreover, it was shown in [EKo1] that the operator defined in this way satisfies (2.3). Without reproducing the proof here we note that it relies on the identities…”

“…In this case the codimension is two and the operator has to be defined by means of boundary conditions involving generalized boundary values as in [EKo1]. We will be able to demonstrate that such a Hamiltonian has a purely absolutely continuous spectrum, and moreover, that its negative part has band structure with an at most finite number of gaps.…”

“…Here and in all the following we choose the principal branch of the logarithm on C \ (−∞, 0]. (Note that we have changed the sign in the definition of T (z) as compared to [EKo1].) We write R 0 (z) := (H 0 − zI) −1 .…”

Hiatus perturbation for a singular Schrödinger operator with an interaction supported by a curve in R3

“…The extension is by far not trivial since the codimension of the singular interaction support influences properties of such Schrödinger operators substantially [AGHH]. In our particular case we know that to define such a Hamiltonian for a curve in R 3 one cannot use, in contrast to the codimension one case, the "natural" quadratic form and has to resort to appropriate generalized boundary conditions [EK02].…”

Schrödinger operators with singular interactions: a model of tunnelling resonances

“…Operators of the type (1.1) or similar have been studied recently with the aim to describe nanostructures which are "leaky" in the sense that they do not neglect quantum tunneling -cf. [7,8,9,10,11,12,13] and references therein. In this sense we can regard the present model with d = 2 as an idealized description of a quantum wire and a collection of quantum dots which are spatially separated but close enough to each other so that electrons are able to pass through the classically forbidden zone separating them.…”