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 $\epsilon$-tubular neighbourhood of a curve in ${\mathbb{R}}^3$ and the object of interest is the Dirichlet Laplacian on this tube in the asymptotic limit $\epsilon \to 0$. We generalise this by considering fibre bundles $M$ over a $d$-dimensional submanifold $B\subset {\mathbb{R}}^{d+k}$ with fibres diffeomorphic to $F\subset {\mathbb{R}}^k$, whose total space is embedded into an $\epsilon$-neighbourhood of $B$. From this point of view $B$ takes the role of the curve and $F$ that of the disc-shaped 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 ${\mathbb{R}}^3$.

References in corpus (3)

• Nonadiabatic couplings and gauge-theoretical structure of curved quantum waveguides
• On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes
• Quantum dynamics of a particle constrained to lie on a surface

Cited by in corpus (11)

• Quantum mechanics of a constrained particle and the problem of prescribed geometry-induced potential
• An effective Hamiltonian for the eigenvalue asymptotics of a Robin Laplacian with a large parameter
• Moving frames and the characterization of curves that lie on a surface
• Spectral properties of relativistic quantum waveguides
• Soft quantum waveguides with an explicit cut-locus
• The adiabatic limit of Schrödinger operators on fibre bundles
• The NLS limit for bosons in a quantum waveguide
• Schrödinger equation in moving domains
• Ruled strips with asymptotically diverging twisting
• Quantum waveguides with magnetic fields
• Fermi's golden rule in tunneling models with quantum waveguides perturbed by Kato class measures