Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator

Originally motivated by a stability problem in Fluid Mechanics, we study the spectral and pseudospectral properties of the differential operator $H_\epsilon = -\partial_x^2 + x^2 + i\epsilon^{-1}f(x)$ on $L^2(R)$, where $f$ is a real-valued function and $\epsilon > 0$ a small parameter. We define $\Sigma(\epsilon)$ as the infimum of the real part of the spectrum of $H_\epsilon$, and $\Psi(\epsilon)^{-1}$ as the supremum of the norm of the resolvent of $H_\epsilon$ along the imaginary axis. Under appropriate conditions on $f$, we show that both quantities $\Sigma(\epsilon)$, $\Psi(\epsilon)$ go to infinity as $\epsilon \to 0$, and we give precise estimates of the growth rate of $\Psi(\epsilon)$. We also provide an example where $\Sigma(\epsilon)$ is much larger than $\Psi(\epsilon)$ if $\epsilon$ is small. Our main results are established using variational "hypocoercive" methods, localization techniques and semiclassical subelliptic estimates.Comment: 38 pages, 4 figure

Three-dimensional stability of Burgers vortices

Burgers vortices are explicit stationary solutions of the Navier-Stokes equations which are often used to describe the vortex tubes observed in numerical simulations of three-dimensional turbulence. In this model, the velocity field is a two-dimensional perturbation of a linear straining flow with axial symmetry. The only free parameter is the Reynolds number $Re = \Gamma/\nu$, where $\Gamma$ is the total circulation of the vortex and $\nu$ is the kinematic viscosity. The purpose of this paper is to show that Burgers vortex is asymptotically stable with respect to general three-dimensional perturbations, for all values of the Reynolds number. This definitive result subsumes earlier studies by various authors, which were either restricted to small Reynolds numbers or to two-dimensional perturbations. Our proof relies on the crucial observation that the linearized operator at Burgers vortex has a simple and very specific dependence upon the axial variable. This allows to reduce the full linearized equations to a vectorial two-dimensional problem, which can be treated using an extension of the techniques developped in earlier works. Although Burgers vortices are found to be stable for all Reynolds numbers, the proof indicates that perturbations may undergo an important transient amplification if $Re$ is large, a phenomenon that was indeed observed in numerical simulations.Comment: 31 pages, no figur

Clinicians' perspectives on the duty of candour: Implications for medical ethics education

© The Author(s) 2017. Content: Truth-telling is an integral part of medical practice in many parts of the world. However, recent public inquiries, including the Francis Inquiry reveal that a duty of candour in practise, are at times compromised. Consequently, the duty of candour became a statutory requirement in England. This study aimed to explore clinicians’ perspectives of the implications of the legislation for medical ethics education, as raising standards to improve patient safety remains an international concern. Methods: One-to-one interviews with clinical educators from various specialties who contribute to the MBChB programme at the authors’ university. Once data saturation had been assessed, transcripts were analysed using a thematic approach by the following concurrent activities: data reduction and coding into themes. Example quotations are used to illustrate that key themes are grounded in the data. Results: Eleven clinical educators were interviewed; three general practitioners, six physicians and two surgeons. Thematic analysis identified three key themes; reaction to legislation, barriers to implementation and areas of the medical curriculum that can be further developed to better prepare future doctors. Conclusions: Currently, the legislation is not reaching all frontline staff; there remains a lack of appropriate training and teaching on the legislation that responds to the perceived challenges to implementing candour. These challenges include tensions in the clinical workplace and concerns about the patient’s best interests conflicting with requirements of the legislation. Both undergraduate and postgraduate curricula need to integrate teaching on the implications of the legislation and take a practice based approach in doing so

Asymptotic diophantine approximation:the multiplicative case

Let $\alpha$ and $\beta$ be irrational real numbers and 0<\F<1/30. We prove a precise estimate for the number of positive integers $q\leq Q$ that satisfy \|q\alpha\|\cdot\|q\beta\|<\F. If we choose \F as a function of $Q$ we get asymptotics as $Q$ gets large, provided \F Q grows quickly enough in terms of the (multiplicative) Diophantine type of $(\alpha,\beta)$, e.g., if $(\alpha,\beta)$ is a counterexample to Littlewood's conjecture then we only need that \F Q tends to infinity. Our result yields a new upper bound on sums of reciprocals of products of fractional parts, and sheds some light on a recent question of L\^{e} and Vaaler.Comment: To appear in Ramanujan Journa

Absolute frequency measurements of 85Rb nF7/2 Rydberg states using purely optical detection

A three-step laser excitation scheme is used to make absolute frequency measurements of highly excited nF7/2 Rydberg states in 85Rb for principal quantum numbers n=33-100. This work demonstrates the first absolute frequency measurements of rubidium Rydberg levels using a purely optical detection scheme. The Rydberg states are excited in a heated Rb vapour cell and Doppler free signals are detected via purely optical means. All of the frequency measurements are made using a wavemeter which is calibrated against a GPS disciplined self-referenced optical frequency comb. We find that the measured levels have a very high frequency stability, and are especially robust to electric fields. The apparatus has allowed measurements of the states to an accuracy of 8.0MHz. The new measurements are analysed by extracting the modified Rydberg-Ritz series parameters.Comment: 12 pages, 5 figures, submitted to New. J. Phy

Dipole Blockade and Quantum Information Processing in Mesoscopic Atomic Ensembles

We describe a technique for manipulating quantum information stored in collective states of mesoscopic ensembles. Quantum processing is accomplished by optical excitation into states with strong dipole-dipole interactions. The resulting ``dipole blockade'' can be used to inhibit transitions into all but singly excited collective states. This can be employed for a controlled generation of collective atomic spin states as well as non-classical photonic states and for scalable quantum logic gates. An example involving a cold Rydberg gas is analyzed

Superconducting proximity effect in clean ferromagnetic layers

We investigate superconducting proximity effect in clean ferromagnetic layers with rough boundaries. The subgap density of states is formed by Andreev bound states at energies which depend on trajectory length and the ferromagnetic exchange field. At energies above the gap, the spectrum is governed by resonant scattering states. The resulting density of states, measurable by tunneling spectroscopy, exhibits a rich structure, which allows to connect the theoretical parameters from experiments.Comment: 11 pages, 5 figures (included

Absolute spin-valve effect with superconducting proximity structures

We investigate spin dependent transport in hybrid superconductor(S)--normal-metal(N)--ferromagnet(F) structures under conditions of proximity effect. We demonstrate the feasibility of the absolute spin-valve effect for a certain interval of voltages in a system consisting of two coupled tri-layer structures. Our results are also valid for non-collinear magnetic configurations of the ferromagnets.Comment: 1 TEX file, 2 Postscript files. Accepted for publication in Physical Review Letter