Operateurs de Schrodinger quasi-periodiques adiabatiques : Interactions entre les bandes spectrales d'un operateur periodique

Study of fine spectral properties of quasiperiodic and similar discrete Schr\"odinger operators involves dealing with problems caused by small denominators, and until recently was only possible using perturbative methods, requiring certain small parameters and complicated KAM-type schemes. We review the recently developed nonperturbative methods for such study which lead to stronger results and are significantly simpler. Numerous applications mainly due to J. Bourgain, M. Gol...

In this paper, we consider Schr\"odinger operators on $L^2(0,\infty)$ given by \begin{align} Hu=(H_0+V)u=-u^{\prime\prime}+V_0u+Vu=Eu,\nonumber \end{align} where $V_0$ is real, $1$-periodic and $V$ is the perturbation. It is well known that under perturbations $V(x)=o(1)$ as $x\to\infty$, the essential spectrum of $H$ coincides with the essential spectrum of $H_0$. We introduce a new way to construct $C^\infty$ oscillatory decaying perturbations. In particular, we can...

We consider a periodic system of domains coupled by small windows. In such domain we study the band spectrum of a Schroedinger operator subject to Neumann condition. We show that near each isolated eigenvalue of the similar operator but in the periodicity cell, there are several non-intersecting bands of the spectrum for the perturbed operator. We also discuss the position of the points at which the band functions attain the edges of each band.

By using quasi--derivatives, we develop a Fourier method for studying the spectral properties of one dimensional Schr\"odinger operators with periodic singular potentials.

In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schr\"odinger operators whose potentials are obtained by real-valued sampling along the orbits of an ergodic invertible transformation. After an introductory part explaining basic spectral concepts and fundamental results, we present the general theory of such operators, and then provide an overview of known results for specific classes of potentials. Here we focus primarily on the...

Schr\"odinger operators with periodic (possibly complex-valued) potentials and discrete periodic operators (possibly with complex-valued entries) are considered, and in both cases the computational spectral problem is investigated: namely, under what conditions can a "one-size-fits-all" algorithm for computing their spectra be devised? It is shown that for periodic banded matrices this can be done, as well as for Schr\"odinger operators with periodic potentials that are suffi...

The spectra of the Schr\"odinger operators with periodic potentials are studied. When the potential is real and periodic, the spectrum consists of at most countably many line segments (energy bands) on the real line, while when the potential is complex and periodic, the spectrum consists of at most countably many analytic arcs in the complex plane. In some recent papers, such operators with complex $\mathcal{PT}$-symmetric periodic potentials are studied. In particular, the...

We discuss the band-gap structure and the integrated density of states for periodic elliptic operators in the Hilbert space $L_2(\R^m)$, for $m \ge 2$. We specifically consider situations where high contrast in the coefficients leads to weak coupling between the period cells. Weak coupling of periodic systems frequently produces spectral gaps or spectral concentration. Our examples include Schr\"odinger operators, elliptic operators in divergence form, Laplace-Beltrami-oper...

We study multi-frequency quasi-periodic Schr\"odinger operators on $\mathbb{Z}$ in the regime of positive Lyapunov exponent and for general analytic potentials. Combining Bourgain's semi-algebraic elimination of multiple resonances with the method of elimination of double resonances via resultants, we establish exponential finite-volume localization as well as the separation between the eigenvalues. In a follow-up paper we develop the method further to show that for potential...

We consider the 1D Schr\"odinger operator $Hy=-y''+(p+q)y$ with a periodic potential $p$ plus compactly supported potential $q$ on the real line. The spectrum of $H$ consists of an absolutely continuous part plus a finite number of simple eigenvalues in each spectral gap $\g_n\ne \es, n\geq 0$, where $\g_0$ is unbounded gap. We prove the following results: 1) we determine the distribution of resonances in the disk with large radius, 2) a forbidden domain for the resonance...