Topologically non-trivial quantum layers

Given a complete non-compact surface embedded in R^3, we consider the Dirichlet Laplacian in a layer of constant width about the surface. Using an intrinsic approach to the layer geometry, we generalise the spectral results of an original paper by Duclos et al. to the situation when the surface does not possess poles. This enables us to consider topologically more complicated layers and state new spectral results. In particular, we are interested in layers built over surfaces with handles or several cylindrically symmetric ends. We also discuss more general regions obtained by compact deformations of certain layers.Comment: 15 pages, 6 figure

The dynamics of 1D Bloch electrons in constant electric fields

We study the dynamics of a 1D Bloch electron subjected to a constant electric field. The periodic potential is supposed to be less singular than the $\delta$-like potential (Dirac comb). We give a rigorous proof of Ao's result \cite{Ao} that for a large class of initial conditions (high momentum regime) there is no localization in momentum space. The proof is based on the mathematical substantiation of the two simplifying assumptions made in physical literature: the transitions between far away bands can be neglected and the transitions at the quasi-crossing can be described by Landau-Zener like formulae. Using the connection between the above model and the driven quantum ring (DQR) shown by Avron and Nemirovski \cite{AvN}, our results imply the increase of energy for weakly singular such DQR and appropiate initial conditions.Comment: 51 pages, 1 figur

Weakly regular Floquet Hamiltonians with pure point spectrum

We study the Floquet Hamiltonian: -i omega d/dt + H + V(t) as depending on the parameter omega. We assume that the spectrum of H is discrete, {h_m (m = 1..infinity)}, with h_m of multiplicity M_m. and that V is an Hermitian operator, 2pi-periodic in t. Let J > 0 and set Omega_0 = [8J/9,9J/8]. Suppose that for some sigma > 0: sum_{m,n such that h_m > h_n} mu_{mn}(h_m - h_n)^(-sigma) < infinity where mu_{mn} = sqrt(min{M_m,M_n)) M_m M_n. We show that in that case there exist a suitable norm to measure the regularity of V, denoted epsilon, and positive constants, epsilon_* & delta_*, such that: if epsilon |Omega_0| - delta_* epsilon and the Floquet Hamiltonian has a pure point spectrum for all omega in Omega_infinity.Comment: 35 pages, Latex with AmsAr

Bound states and scattering in quantum waveguides coupled laterally through a boundary window

We consider a pair of parallel straight quantum waveguides coupled laterally through a window of a width $\ell$ in the common boundary. We show that such a system has at least one bound state for any $\ell>0$. We find the corresponding eigenvalues and eigenfunctions numerically using the mode--matching method, and discuss their behavior in several situations. We also discuss the scattering problem in this setup, in particular, the turbulent behavior of the probability flow associated with resonances. The level and phase--shift spacing statistics shows that in distinction to closed pseudo--integrable billiards, the present system is essentially non--chaotic. Finally, we illustrate time evolution of wave packets in the present model.Comment: LaTeX text file with 12 ps figure

Limiting absorption principle and perfectly matched layer method for Dirichlet Laplacians in quasi-cylindrical domains

We establish a limiting absorption principle for Dirichlet Laplacians in quasi-cylindrical domains. Outside a bounded set these domains can be transformed onto a semi-cylinder by suitable diffeomorphisms. Dirichlet Laplacians model quantum or acoustically-soft waveguides associated with quasi-cylindrical domains. We construct a uniquely solvable problem with perfectly matched layers of finite length. We prove that solutions of the latter problem approximate outgoing or incoming solutions with an error that exponentially tends to zero as the length of layers tends to infinity. Outgoing and incoming solutions are characterized by means of the limiting absorption principle.Comment: to appear in SIAM Journal on Mathematical Analysi