Semiclassical Resonances of Schr\"odinger operators as zeroes of regularized determinants

We prove that the resonances of long range perturbations of the (semiclassical) Laplacian are the zeroes of natural perturbation determinants. We more precisely obtain factorizations of these determinants of the form $\prod_{w = {\rm resonances}}(z-w) \exp (\varphi_p(z,h))$ and give semiclassical bounds on $\partial_z \varphi_p$ as well as a representation of Koplienko's regularized spectral shift function. Here the index $p \geq 1$ depends on the decay rate at infinity of the perturbation.Comment: 37 pages, published versio

Convergence of a Vector Penalty Projection Scheme for the Navier-Stokes Equations with moving body

In this paper, we analyse a Vector Penalty Projection Scheme (see [1]) to treat the displacement of a moving body in incompressible viscous flows in the case where the interaction of the fluid on the body can be neglected. The presence of the obstacle inside the computational domain is treated with a penalization method introducing a parameter $\eta$. We show the stability of the scheme and that the pressure and velocity converge towards a limit when the penalty parameter $\epsilon$, which induces a small divergence and the time step $\delta$t tend to zero with a proportionality constraint $\epsilon$ = $\lambda$$\delta$t. Finally, when $\eta$ goes to 0, we show that the problem admits a weak limit which is a weak solution of the Navier-Stokes equations with no-sleep condition on the solid boundary. R{\'e}sum{\'e} Dans ce travail nous analysons un sch{\'e}ma de projection vectorielle (voir [1]) pour traiter le d{\'e}placement d'un corps solide dans un fluide visqueux incompressible dans le cas o` u l'interaction du fluide sur le solide est n{\'e}gligeable. La pr{\'e}sence de l'obstacle dans le domaine solide est mod{\'e}lis{\'e}e par une m{\'e}thode de p{\'e}nalisation. Nous montrons la stabilit{\'e} du sch{\'e}ma et la convergence des variables vitesse-pression vers une limite quand le param etre $\epsilon$ qui assure une faible divergence et le pas de temps $\delta$t tendent vers 0 avec une contrainte de proportionalit{\'e} $\epsilon$ = $\lambda$$\delta$t. Finalement nous montrons que leprob{\`i} eme converge au sens faible vers une solution des equations de Navier-Stokes avec une condition aux limites de non glissement sur lafront{\`i} ere immerg{\'e}e quand le param etre de p{\'e}nalisation $\eta$ tend vers 0

Discrete Spectrum of Quantum Hall Effect Hamiltonians I. Monotone Edge Potential

We consider the unperturbed operator $H_0 : = (-i \nabla - A)^2 + W$, self-adjoint in $L^2(\R^2)$. Here $A$ is a magnetic potential which generates a constant magnetic field $b>0$, and the edge potential $W$ is a non-decreasing non constant bounded function depending only on the first coordinate $x \in \R$ of $(x,y) \in \R^2$. Then the spectrum of $H_0$ has a band structure and is absolutely continuous; moreover, the assumption $\lim_{x \to \infty}(W(x) - W(-x)) < 2b$ implies the existence of infinitely many spectral gaps for $H_0$. We consider the perturbed operators $H_{\pm} = H_0 \pm V$ where the electric potential $V \in L^{\infty}(\R^2)$ is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of $H_\pm$ in the spectral gaps of $H_0$. We introduce an effective Hamiltonian which governs the main asymptotic term; this Hamiltonian involves a pseudo-differential operator with generalized anti-Wick symbol equal to $V$. Further, we restrict our attention on perturbations $V$ of compact support and constant sign. We establish a geometric condition on the support of $V$ which guarantees the finiteness of the eigenvalues of $H_{\pm}$ in any spectral gap of $H_0$. In the case where this condition is violated, we show that, generically, the convergence of the infinite series of eigenvalues of $H_+$ (resp. $H_-$) to the left (resp. right) edge of a given spectral gap, is Gaussian.Comment: 32 page

The ethical imperatives of the COVID 19 pandemic: a review from data ethics

In this review, we present some ethical imperatives observed in this pandemic from a data ethics perspective. Our exposition connects recurrent ethical problems in the discipline, such as, privacy, surveillance, transparency, accountability, and trust, to broader societal concerns about equality, discrimination, and justice. We acknowledge data ethics role as significant to develop technological, inclusive, and pluralist societies. - - - Resumen: En esta revisión, exponemos algunos de los imperativos éticos observados desde la ética de datos en esta pandemia. Nuestra exposición busca conectar problemas éticos típicos dentro de esta disciplina, a saber, privacidad, vigilancia, transparencia, responsabilidad y confianza, con preocupaciones a nivel social relacionadas con la igualdad, discrimi nación y justicia. Consideramos que la ética de datos tiene un rol significativo para desarrollar sociedades tecnificadas, inclusivas y pluralistas

ON THE GROUND STATE ENERGY OF THE LAPLACIAN WITH A MAGNETIC FIELD CREATED BY A RECTILINEAR CURRENT

ABSTRACT. We consider the three-dimensional Laplacian with a magnetic field created by an infinite rectilinear current bearing a constant current. The spectrum of the associated Hamiltonian is the positive half-axis as the range of an infinity of band functions all decreasing toward 0. We make a precise asymptotics of the band functions near the ground state energy and we exhibit a semi-classical behavior. We perturb the Hamiltonian by an electric potential. Helped by the analysis of the band functions, we show that for slow decaying potential an infinite number of negative eigenvalues are created whereas only finite number of eigenvalues appears for fast decaying potential. The criterion about finiteness depends essentially on the decay rate of the potential with respect to the distance to the wire