Almost-periodic Potential Research Articles

Invariant tori of full dimension for a nonlinear Schrödinger equation with an almost periodic potential

In this paper, we study a nonlinear Schrödinger equation with an almost-periodic potential, subject to periodic boundary conditions. Under appropriate assumptions on the potential, we obtain many small-amplitude, time-almost-periodic solutions for the equation. We improve the Kolmogorov–Arnold–Moser iteration to reduce the almost-periodic potential, especially depending on the space variables. The analysis of the nonlinear perturbation is also the key argument.

Perturbation Theory for Almost-Periodic Potentials I: One-Dimensional Case

We consider the family of operators {H^{(varepsilon)}:=-frac{d^2}{dx^2}+varepsilon V} in {mathbb{R}} with almost-periodic potential V. We study the behaviour of the integrated density of states (IDS) {N(H^{(varepsilon)};lambda)} when {varepsilonto 0} and {lambda} is a fixed energy. When V is quasi-periodic (i.e. is a finite sum of complex exponentials), we prove that for each {lambda} the IDS has a complete asymptotic expansion in powers of {varepsilon}; these powers are either integer, or in some special cases half-integer. These results are new even for periodic V. We also prove that when the potential is neither periodic nor quasi-periodic, there is an exceptional set {mathcal{S}} of energies (which we call the super-resonance set) such that for any {sqrtlambdanotinmathcal{S}} there is a complete power asymptotic expansion of IDS, and when {sqrtlambdainmathcal{S}}, then even two-terms power asymptotic expansion does not exist. We also show that the super-resonant set {mathcal{S}} is uncountable, but has measure zero. Finally, we prove that the length of any spectral gap of {H^{(varepsilon)}} has a complete asymptotic expansion in natural powers of {varepsilon} when {varepsilon to 0}.

Destabilization of a vortex by acoustic waves

The linear stability of a circular vortex interacting with two plane acoustic waves propagating in opposite directions is investigated. When the wavelength is large compared to the size of the vortex, the core is subjected to time-periodic compressions and strains. A stability analysis is performed with the geometrical optics approximation, which considers short-wavelength perturbations evolving along the trajectories of the basic flow. On the vortex core, the problem is reduced to a single Hill–Schrödinger equation with periodic or almost-periodic potential, the solution to which grows exponentially when parametric resonances occur. On interacting with the acoustic waves, the circular vortex is thus unstable to three-dimensional perturbations.

Electron localization in quasicrystals.

The question of whether scattering by the almost-periodic lattice potentials of recently discovered metallic quasicrystals is able to explain the recently observed short mean free paths (about 1 \AA{}) in these materials is discussed. It is shown that scattering by a three-dimensional, almost-periodic potential cannot account for such resistivities, but a model which includes structural defects in the three-dimensional Penrose tiling and large scattering of electrons due to $s\ensuremath{-}d$ resonant scattering can account for the experimental observations.

Reconstruction of the function of the rotation number for the Schr�dinger operator with almost-periodic potential from a countable set of polynomial invariance laws

Reconstruction of the function of the rotation number for the Schr�dinger operator with almost-periodic potential from a countable set of polynomial invariance laws

ON THE CLOSURE OF THE SET OF FINITE-ZONE POTENTIALS

A special class of almost-periodic potentials of the one-dimensional Schr?dinger operator is studied. These potentials are limits of finite-zone potentials. In contrast to previous work of the author in which a class of almost-periodic potentials was considered with lacunae in the spectrum having a single limit point at infinity, in this paper the case is studied where the lacunae in the spectrum also have finite limit points. There may be infinitely many of the latter.Figures: 2. Bibliography: 10 titles.

Structure of the spectrum of the Schr�dinger operator with almost-periodic potential in the vicinity of its left edge

Structure of the spectrum of the Schr�dinger operator with almost-periodic potential in the vicinity of its left edge

Electrons in random and almost-periodic potentials

A review of results concerning the problem of electrons in random or almost-periodic potentials is given with special emphasis on rigorous results; some more speculative discussions about the upper critical dimension of the localization problem is however developed at the end. Applications of similar ideas to other problems of wave propagation in inhomogeneous media are also mentioned.

Chaotic band structure of almost-periodic potentials

It has been conjectured that the band structure of an almost-periodic potential is a Cantor set. Numerical calculations are presented which indeed suggest that there can be an infinite number of allowed and forbidden bands, in a finite energy range.

The spectrum of a Sturm ? Liouville operator with almost-periodic potential

The spectrum of a Sturm ? Liouville operator with almost-periodic potential