Geometrically induced spectrum in curved leaky wires

Abstract

We study measure perturbations of the Laplacian in L 2 (R 2 ) supported by an infinite curve Γ in the plane which is asymptotically straight in a suitable sense. We show that if Γ is not a straight line, such a "leaky quantum wire" has at least one bound state below the threshold of the essential spectrum.

On Critical Stability of Three Quantum Charges Interacting Through Delta Potentials

“…It has been shown in [1] and [2] that these delta models serve as effective Hamiltonians for atoms in intense magnetic fields or quasi-particles in carbon nanotubes. As one can see in ([4], [5], [6], [3]), they also seem to be relevant for atomic wave guides, nano and leaky wires.…”

Gap asymptotics in a weakly bent leaky quantum wire

“…Since the curve Γ belongs under the assumptions we made into the class analyzed in Ref. [10], we know that 4) and that H Γ has at least one discrete eigenvalue whenever Γ = Σ.…”

“…If it is a straight line, Γ = Σ, the spectrum is easily found by separation of variables: it is absolutely continuous and σ(H Σ ) = − Once the curve becomes geometrically nontrivial, the spectrum changes. In particular, if Γ is asymptotically straight at both end in an appropriate sense, then the essential spectrum remain preserved, σ ess (H Γ ) = − eigenvalues appear below its threshold [10]. Relations between between properties of this discrete spectrum and the geometry of the curve are of a great interest.…”

“…In the waveguide case the eigenvalue existence is proved variationally and the asymptotics uses the Birman-Schwinger method adapted from the standard Schrödinger operator theory. Here the (generalized) BirmanSchwinger trick was used already to prove the existence [10] and the only natural way to proceed is to analyze finer properties of the BS operator. Nevertheless, our main goal in this paper is to demonstrate that the mentioned conjecture was correct, namely that we have the asymptotic relation…”

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.…”