On kernel engineering via Paley–Wiener

A radial basis function approximation takes the form $$s(x)=\sum_{k=1}^na_k\phi(x-b_k),\quad x\in {\mathbb{R}}^d,$$ where the coefficients a 1,…,a n are real numbers, the centres b 1,…,b n are distinct points in ℝ d , and the function φ:ℝ d →ℝ is radially symmetric. Such functions are highly useful in practice and enjoy many beautiful theoretical properties. In particular, much work has been devoted to the polyharmonic radial basis functions, for which φ is the fundamental solution of some iterate of the Laplacian. In this note, we consider the construction of a rotation-invariant signed (Borel) measure μ for which the convolution ψ=μ φ is a function of compact support, and when φ is polyharmonic. The novelty of this construction is its use of the Paley–Wiener theorem to identify compact support via analysis of the Fourier transform of the new kernel ψ, so providing a new form of kernel engineering

Quantum tunneling through planar p-n junctions in HgTe quantum wells

We demonstrate that a p-n junction created electrically in HgTe quantum wells with inverted band-structure exhibits interesting intraband and interband tunneling processes. We find a perfect intraband transmission for electrons injected perpendicularly to the interface of the p-n junction. The opacity and transparency of electrons through the p-n junction can be tuned by changing the incidence angle, the Fermi energy and the strength of the Rashba spin-orbit interaction. The occurrence of a conductance plateau due to the formation of topological edge states in a quasi-one-dimensional p-n junction can be switched on and off by tuning the gate voltage. The spin orientation can be substantially rotated when the samples exhibit a moderately strong Rashba spin-orbit interaction.Comment: 4 pages, 4 figure

Low-field diffusion magneto-thermopower of a high mobility two-dimensional electron gas

The low magnetic field diffusion thermopower of a high mobility GaAs-heterostructure has been measured directly on an electrostatically defined micron-scale Hall-bar structure at low temperature (T = 1.6 K) in the low magnetic field regime (B < 1.2 T) where delocalized quantum Hall states do not influence the measurements. The sample design allowed the determination of the field dependence of the thermopower both parallel and perpendicular to the temperature gradient, denoted respectively by Sxx (longitudinal thermopower) and Syx (Nernst-Ettinghausen coefficient). The experimental data show clear oscillations in Sxx and Syx due to the formation of Landau levels for 0.3 T < B < 1.2 T and reveal that Syx is approximately 120 times larger than Sxx at a magnetic field of 1 T, which agrees well with the theoretical prediction.Comment: 4 pages, 4 figure

Casimir force on amplifying bodies

Based on a unified approach to macroscopic QED that allows for the inclusion of amplification in a limited space and frequency range, we study the Casimir force as a Lorentz force on an arbitrary partially amplifying system of linearly locally responding (isotropic) magnetoelectric bodies. We demonstrate that the force on a weakly polarisable/magnetisable amplifying object in the presence of a purely absorbing environment can be expressed as a sum over the Casimir--Polder forces on the excited atoms inside the body. As an example, the resonant force between a plate consisting of a dilute gas of excited atoms and a perfect mirror is calculated

Van-der-Waals potentials of paramagnetic atoms

We study single- and two-atom van der Waals interactions of ground-state atoms which are both polarizable and paramagnetizable in the presence of magneto-electric bodies within the framework of macroscopic quantum electrodynamics. Starting from an interaction Hamiltonian that includes particle spins, we use leading-order perturbation theory for the van der Waals potentials expressed in terms of the polarizability and magnetizability of the atom(s). To allow for atoms embedded in media, we also include local-field corrections via the real-cavity model. The general theory is applied to the potential of a single atom near a half space and that of two atoms embedded in a bulk medium or placed near a sphere, respectively.Comment: 18 pages, 3 figures, 1 tabl

An Anderson-Fano Resonance and Shake-Up Processes in the Magneto-Photoluminescence of a Two-Dimensional Electron System

We report an anomalous doublet structure and low-energy satellite in the magneto-photoluminescence spectra of a two-dimensional electron system. The doublet structure moves to higher energy with increasing magnetic field and is most prominent at odd filling factors 5 and 3. The lower-energy satellite peak tunes to lower energy for increasing magnetic field between filling factor 6 and 2. These features occur at energies below the fundamental band of recombination originating from the lowest Landau level and display striking magnetic field and temperature dependence that indicates a many-body origin. Drawing on a recent theoretical description of Hawrylak and Potemski, we show that distinct mechanisms are responsible for each feature.Comment: 14 pages including 5 figures. To appear in the April 15th edition of Phy. Rev. B. rapid com

Counting approximately-shortest paths in directed acyclic graphs

Given a directed acyclic graph with positive edge-weights, two vertices s and t, and a threshold-weight L, we present a fully-polynomial time approximation-scheme for the problem of counting the s-t paths of length at most L. We extend the algorithm for the case of two (or more) instances of the same problem. That is, given two graphs that have the same vertices and edges and differ only in edge-weights, and given two threshold-weights L_1 and L_2, we show how to approximately count the s-t paths that have length at most L_1 in the first graph and length at most L_2 in the second graph. We believe that our algorithms should find application in counting approximate solutions of related optimization problems, where finding an (optimum) solution can be reduced to the computation of a shortest path in a purpose-built auxiliary graph

Nonlocal edge state transport in the quantum spin Hall state

We present direct experimental evidence for nonlocal transport in HgTe quantum wells in the quantum spin Hall regime, in the absence of any external magnetic field. The data conclusively show that the non-dissipative quantum transport occurs through edge channels, while the contacts lead to equilibration between the counter-propagating spin states at the edge. We show that the experimental data agree quantitatively with the theory of the quantum spin Hall effect.Comment: 13 pages, 4 figure