Asymptotic Analysis

Here we give results on trace asymptotics with small remainder estimates. We treat situations where the spectral parameter is implicit, and where there is no really natural associated evolution equation. We apply these results to a periodic Schrödinger operator with two different types of perturbations: slowly varying and strong. In both cases we get precise remainder estimates for the counting function of eigenvalues of the perturbed periodic Schrödinger operator in a gap of the non‐perturbed one.

The limiting behavior of the unique solution to Maxwell equations in an exterior domain with a Leontovich boundary condition as the impedance tends to zero is investigated. This is accomplished by reducing the impedance boundary value problem for the Maxwell equations to a system of three integral equations. It is shown that the Leontovich boundary condition leads to a singular perturbation problem for the Maxwell equations. A specific numerical treatment is required to achieve a sufficient accuracy.

We consider the problem of minimizing the bending energy $E_{\rm b}=\int\kappa^2\,{\rm d}s$ on isotopy classes of closed curves in $\mathbb{R}^3$ to model the elastic behaviour of knotted loops of springy wire. A potential of Coulomb type with a small factor $\theta$ as a measure for the thickness of the wire is added to the elastic energy in order to preserve the isotopy class. With a direct method we show existence of minimizers $\mathbf{x }^{\theta }$ under a given topological knot type for each $\theta>0$ . Moreover, allowing smaller and smaller thickness ( $\theta\searrow0$ ) and looking at a subsequence of the corresponding minimizers $\mathbf{x }^{\theta}$ , we obtain a generalized minimizer $\mathbf{x }$ of the bending energy $E_{\rm b}$ as a limit. It turns out that $\mathbf{x }$ is the once covered circle, if one considers the class of unknotted loops in ${\Bbb R}^3$ . In nontrivial knot classes, however, $\mathbf{x }$ must have double points, whose multiplicity and position on the curve is controlled by the value of the bending energy $E_{\rm b}(\mathbf{x })$ .

The transient drift diffusion equations describing the charge transport in semiconductors coupled with Maxwell’s equations are considered. The asymptotic behavior of the solutions is investigated. In particular it is shown that they converge for $t\rightarrow\infty$ to solutions of the stationary drift diffusion equations provided that the currents corresponding to the stationary states are sufficiently small.

Oscillating approximate solutions to nonlinear hyperbolic dispersive systems are studied. Ansatz of three scales are used in order to deal with diffractive effects. The scaling of the approximate solutions is chosen so that diffractive, dispersive effects and rectification are present in the leading term.

The propagation along the rays of geometrical optics of the oscillating Fourier coefficients of the leading terms is corrected by a Schrödinger dispersion which appears for long times only. The propagation of the nonoscillating Fourier coefficient depends on the properties of a symmetric hyperbolic system, whose characteristic variety is the tangent cone at $0$ to the characteristic variety of the initial operator.

Equations determining the leading term require a sublinear growth condition for the corrector and the inroduction of the analytical “average operators” which convey this sublinear growth condition in a simple way and sort the nonlinearities out.

In the last part, detailed physical examples are given.

In this paper, we study the inverse scattering related to a channel scattering operator $S_{\alpha\beta}$ in $N$ ‐body problems. Given the incoming and outgoing channels $\alpha$ , $\beta$ associated with a same, but arbitary cluster decomposition, $a$ , of the $N$ ‐body system, we prove that all effective interactions between the mass‐centers of the clusters in $a$ can be reconstructed from the high energy asymptotics of scattering matrices $S_{\alpha\beta}(\lambda)$ .

In this paper, we discuss the limit behaviour of solutions to a class of boundary value problems with equivalued surface for linear elliptic equations when the equivalued surface boundary shrinks to a fixed point on the boundary.