Within the effective mass and nonparabolic band theory, a general framework of mathematical models and numerical methods is developed for theoretical studies of semiconductor quantum dots. It includes single-electron models and many-electron models of Hartree-Fock, configuration interaction, and current-spin density functional theory approaches. These models result in nonlinear eigenvalue problems from a suitable discretization. Cubic and quintic Jacobi-Davidson methods of block or nonblock version...

We study the time-harmonic acoustic scattering in a duct in presence of a flow and of a discontinuous impedance boundary condition. Unlike a continuous impedance, a discontinuous one leads to still open modeling questions, as in particular the singularity of the solution at the abrupt transition and the choice of the right unknown to formulate the scattering problem. To address these questions we propose a mathematical approach based on variational formulations set in weighted Sobolev spaces. Considering...

The Laplacian ${\Delta }_g$ of a compact Riemannian manifold $(M,g)$ is called maximally degenerate if its eigenvalue multiplicity function $m_g\left(k\right)$ is of maximal growth among metrics of the same dimension and volume. Canonical spheres $(S^n,\mathrm{can})$ and CROSSes are MD, and one asks if they are the only examples. We show that a MD metric must be at least a Zoll metric with just one distinct eigenvalue in each cluster, and hence with all band invariants equal to zero. The principal band invariant is then calculated in terms of geodesic...

Motivated by the fundamental theorem of calculus, and based on the works of W. Feller as well as M. Kac and M. G. Kreĭn, given an atomless Borel probability measure $\eta$ supported on a compact subset of $ℝ$ U. Freiberg and M. Zähle introduced a measure-geometric approach to define a first order differential operator ${\nabla }_{\eta }$ and a second order differential operator ${\Delta }_{\eta }$, with respect to $\eta$. We generalize this approach to measures of the form $\eta :=\nu +\delta$, where $\nu$ is non-atomic and $\delta$ is finitely supported. We determine analytic...

We obtain some microlocal estimates of the resonant states associated to a resonance $z_0$ of an $h$-differential operator. More precisely, we show that the normalized resonant states are $𝒪(\sqrt{|\mathrm{Im}{z}_0|/h}$$+h^{\infty }...$

We consider the variational problem         inf{αλ1(Ω) + βλ2(Ω) + (1 − α − β)λ3(Ω) | Ω open in ℝn, |Ω| ≤ 1}, for α, β ∈ [0, 1], α + β ≤ 1, where λk(Ω) is the kth eigenvalue of the Dirichlet Laplacian acting in L2(Ω) and |Ω| is the Lebesgue measure of Ω. We investigate for which values of α, β every minimiser is connected.

Soit $M$ une variété hyperbolique compacte de dimension 3, de diamètre $d$ et de volume $\le V$. Si on note ${\mu }_i\left(M\right)$ la $i$-ième valeur propre du laplacien de Hodge-de Rham agissant sur les 1-formes coexactes de $M$, on montre que ${\mu }_1\left(M\right)\ge \frac{c}{d^3{e}^{2kd}}$ et ${\mu }_{k+1}\left(M\right)\ge \frac{c}{d^2}$, où $c\>0$ est une constante ne dépendant que de $V$, et $k$ est le nombre de composantes connexes de la partie mince de $M$. En outre, on montre que pour toute 3-variété hyperbolique $M_{\infty }$ de volume fini avec cusps, il existe une suite $M_i$ de remplissages compacts de $M_{\infty }$, de diamètre $d_i\to +\infty$ telle que et ${\mu }_1\left(M_i\right)\ge \frac{c}{d_i^2}$.

We state a superconvergence result for the lowest order Raviart-Thomas approximation of eigenvalue problems. It is known that a similar superconvergence result holds for the mixed approximation of Laplace problem; here we introduce a new proof, since the one given for the source problem cannot be generalized in a straightforward way to the eigenvalue problem. Numerical experiments confirm the superconvergence property and suggest that it also holds for the lowest order Brezzi-Douglas-Marini...