Generalised Quantum Waveguides

Abstract

We study general quantum waveguides and establish explicit effective Hamiltonians for the Laplacian on these spaces. A conventional quantum waveguide is an ε-tubular neighbourhood of a curve in R 3 and the object of interest is the Dirichlet Laplacian on this tube in the asymptotic limit ε → 0. We generalise this by considering fibre bundles M over a d-dimensional submanifold B ⊂ R d+k with fibres diffeomorphic to F ⊂ R k , whose total space is embedded into an ε-neighbourhood of B. From this point of view B takes the role of the curve and F that of the discshaped cross-section of a conventional quantum waveguide. Our approach allows, among other things, for waveguides whose cross-sections F are deformed along B and also the study of the Laplacian on the boundaries of such waveguides. By applying recent results on the adiabatic limit of Schrödinger operators on fibre bundles we show, in particular, that for small energies the dynamics and the spectrum of the Laplacian on M are reflected by the adiabatic approximation associated to the ground state band of the normal Laplacian. We give explicit formulas for the according effective operator on L 2 (B) in various scenarios, thereby improving and extending many of the known results on quantum waveguides and quantum layers in R 3 .

Ruled Strips with Asymptotically Diverging Twisting

“…Analogous results are also known for tubes embedded in a Riemannian manifold A instead of the Euclidean space: see [11,12,19,10,8,18]. In the simplest non-trivial situation where Σ is a curve in a two-dimensional surface A and the cross-section ω is a symmetric interval, any non-trivial curvature of Σ and/or non-negative Gauss curvature of A, both vanishing at infinity in an appropriate sense, lead to the existence of discrete spectra (see [11]), while Hardy-type inequalities hold if Σ is a geodesic and A is non-positively curved (see [12,10]).…”

Quantum waveguides with magnetic fields

“…In this work we study generalised quantum waveguides as introduced in [HLT15] in the presence of moderate and strong external magnetic fields. In a nutshell, a generalised quantum waveguide is an ε-thin neighbourhood of a submanifold, or a boundary thereof.…”

“…There exists a vast literature on specific types of quantum waveguides without magnetic fields, see e.g. [EK15,HLT15] and references therein. There are significantly less results on the magnetic case.…”

“…In Section 2, we identify the family {T ε } 0<ε≤1 (the "tube") with an ε-independent manifold M (the "waveguide") that has the additional structure of a fibre bundle M π M − − → B, with typical fibre ("typical cross-section") F given by a compact subset of R f . The corresponding diffeomorphism Ψ ε : M → T ε is described in [HLT15, Chapter 2] and lifts to a unitary operator L 2 (T ε ) → L 2 (M, vol G ε ).…”

Nonadiabatic couplings and gauge-theoretical structure of curved quantum waveguides

“…Assuming a particular scaling behavior of the various length scales, the first few orders of this adiabatic perturbation theory expansion for the quantum waveguide problem have been worked out in [62,63] (see also the recent work Ref. [64]). While this perturbation scheme is mathematically rigorous and insightful from a formal point of view, it can be expected that in many situations the plain transverse eigenstates will persist to play a crucial role as immediate, intuitive points of reference, also since they are accessible to direct measurements [65][66][67].…”