Magnetic WKB Constructions

This paper is devoted to the semiclassical magnetic Laplacian. Until now WKB expansions for the eigenfunctions were only established in presence of a non-zero electric potential. Here we tackle the pure magnetic case. Thanks to Feynman-Hellmann type formulas and coherent states decomposition, we develop here a magnetic Born-Oppenheimer theory. Exploiting the multiple scales of the problem, we are led to solve an effective eikonal equation in pure magnetic cases and to obtain WKB expansions. We also investigate explicit examples for which we can improve our general theorem: global WKB expansions, quasi-optimal estimates of Agmon and upper bound of the tunelling effect (in symmetric cases). We also apply our strategy to get more accurate descriptions of the eigenvalues and eigenfunctions in a wide range of situations analyzed in the last two decades

Mathematical theory for the interface mode in a waveguide bifurcated from a Dirac point

In this paper, we prove the existence of a bound state in a waveguide that consists of two semi-infinite periodic structures separated by an interface. The two periodic structures are perturbed from the same periodic medium with a Dirac point and they possess a common band gap enclosing the Dirac point. The bound state, which is called interface mode here, decays exponentially away from the interface with a frequency located in the common band gap and can be viewed as a bifurcation from the Dirac point. Using the layer potential technique and asymptotic analysis, we first characterize the band gap opening for the two perturbed periodic media and derive the asymptotics of the Bloch modes near the band gap edges. By formulating the eigenvalue problem for the waveguide with two semi-infinite structures using a boundary integral equation over the interface and analyzing the characteristic values of the associated boundary integral operator, we prove the existence of the interface mode for the waveguide when the perturbation of the periodic medium is small

Asymptotics of the spectrum of the mixed boundary value problem for the Laplace operator in a thin spindle-shaped domain

Free access to full-text articles is allowed 3 years after publication of the corresponding issue. Access to full-text articles of this issue will be allowed starting from April 1, 2024The asymptotics is examined for solutions to the spectral problem for the Laplace operator in a d-dimensional thin, of diameter O(h), spindle-shaped domain Omega(h) with the Dirichlet condition on small, of size h +0, an ordinary differential equation on the axis (-1, 1) (sic) z of the spindle arises with a coefficient degenerating at the points z = +/- 1 and moreover, without any boundary condition because the requirement on the boundedness of eigenfunctions makes the limit spectral problem well-posed. Error estimates are derived for the one-dimensional model but in the case of d = 3 it is necessary to construct boundary layers near the sets Gamma(h)(+/-) and in the case of d = 2 it is necessary to deal with selfadjoint extensions of the differential operator. The extension parameters depend linearly on In h so that its eigenvalues are analytic functions in the variable 1/vertical bar ln h vertical bar As a result, in all dimensions the one-dimensional model gets the power-law accuracy O(h(delta)d ) with an exponent delta(d) > 0. First (the smallest) eigenvalues, positive in Omega(h) and null in (-1, 1), require individual treatment. Also, infinite asymptotic series are discussed, as well as the static problem (without the spectral parameter) and related shapes of thin domains.Peer reviewe

Free Vibrations of Axisymmetric Shells: Parabolic and Elliptic cases

Approximate eigenpairs (quasimodes) of axisymmetric thin elastic domains with laterally clamped boundary conditions (Lam{\'e} system) are determined by an asymptotic analysis as the thickness ($2\varepsilon$) tends to zero. The departing point is the Koiter shell model that we reduce by asymptotic analysis to a scalar modelthat depends on two parameters: the angular frequency $k$ and the half-thickness $\varepsilon$. Optimizing $k$ for each chosen $\varepsilon$, we find power laws for $k$ in function of $\varepsilon$ that provide the smallest eigenvalues of the scalar reductions.Corresponding eigenpairs generate quasimodes for the 3D Lam{\'e} system by means of several reconstruction operators, including boundary layer terms. Numerical experiments demonstrate that in many cases the constructed eigenpair corresponds to the first eigenpair of the Lam{\'e} system.Geometrical conditions are necessary to this approach: The Gaussian curvature has to be nonnegative and the azimuthal curvature has to dominate the meridian curvature in any point of the midsurface. In this case, the first eigenvector admits progressively larger oscillation in the angular variable as $\varepsilon$ tends to $0$. Its angular frequency exhibits a power law relationof the form $k=\gamma \varepsilon^{-\beta}$ with $\beta=\frac14$ in the parabolic case (cylinders and trimmed cones), and the various $\beta$s $\frac25$, $\frac37$, and $\frac13$ in the elliptic case.For these cases where the mathematical analysis is applicable, numerical examples that illustrate the theoretical results are presented