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

In the theory of ergodic one-dimensional Schrodinger operators, ac spectrum has been traditionally expected to be very rigid. Two key conjectures in this direction state, on one hand, that ac spectrum demands almost periodicity of the potential, and, on the other hand, that the eigenfunctions are almost surely bounded in the essential suport of the ac spectrum. We show how the repeated slow deformation of periodic potentials can be used to break rigidity, and disprove both co...

We discover that the distribution of (frequency and phase) resonances plays a role in determining the spectral type of supercritical quasi-periodic Schr\"odinger operators. In particular, we disprove the second spectral transition line conjecture of Jitomirskaya in the early 1990s.

In this article we consider the one-dimensional Schrodinger operator L(Q) with a Hermitian periodic m by m matrix potential Q. We investigate the bands and gaps of the spectrum and prove that the main part of the positive real axis is overlapped by m bands. Moreover, we find a condition on the potential Q for which the number of gaps in the spectrum of L(Q) is finite.

We consider the one-dimensional discrete Schr\"odinger operator $$ \bigl[H(x,\omega)\varphi\bigr](n)\equiv -\varphi(n-1)-\varphi(n+1) + V(x + n\omega)\varphi(n)\ , $$ $n \in \mathbb{Z}$, $x,\omega \in [0, 1]$ with real-analytic potential $V(x)$. Assume $L(E,\omega)>0$ for all $E$. Let $\mathcal{S}_\omega$ be the spectrum of $H(x,\omega)$. For all $\omega$ obeying the Diophantine condition $\omega \in \mathbb{T}_{c,a}$, we show the following: if $\mathcal{S}_\omega \cap (E',E"...

This paper summarizes the contents of a plenary talk given at the 14th Biennial Conference of Indian SIAM in Amritsar in February 2018. We discuss here the effect of an abrupt spectral change for some classes of Schr\"odinger operators depending on the value of the coupling constant, from below bounded and partly or fully discrete, to the continuous one covering the whole real axis. A prototype of such a behavior can be found in Smilansky-Solomyak model devised to illustrate ...

We consider one-dimensional difference Schroedinger equations on the discrete line with a potential generated by evaluating a real-analytic potential function V(x) on the one-dimensional torus along an orbit of the shift x-->x+nw. If the Lyapunov exponent is positive for all energies and w, then the integrated density of states is absolutely continuous for almost every w. In this work we establish the formation of a dense set of gaps in the spectrum. Our approach is based on ...

We show that the spectrum of a discrete two-dimensional periodic Schr\"odinger operator on a square lattice with a sufficiently small potential is an interval, provided the period is odd in at least one dimension. In general, we show that the spectrum may consist of at most two intervals and that a gap may only open at energy zero. This sharpens several results of Kr\"uger and may be thought of as a discrete version of the Bethe--Sommerfeld conjecture. We also describe an app...

We study the quasi-periodic Schr\"odinger equation $$ -\psi"(x) + V(x) \psi(x) = E \psi(x), \qquad x \in \IR $$ in the regime of "small" $V$. Let $(E_m',E"_m)$, $m \in \zv$, be the standard labeled gaps in the spectrum. Our main result says that if $E"_m - E'_m \le \ve \exp(-\kappa_0 |m|)$ for all $m \in \zv$, with $\ve$ being small enough, depending on $\kappa_0 > 0$ and the frequency vector involved, then the Fourier coefficients of $V$ obey $|c(m)| \le \ve^{1/2} \exp(-\fra...

This work is devoted to the study of a family of almost periodic one-dimensional Schr\"odinger equations. We define a monodromy matrix for this family. We study the asymptotic behavior of this matrix in the adiabatic case. Therefore, w develop a complex WKB method for adiabatic perturbations of periodic Schr\"odinger equations. At last, the study of the monodromy matrix enables us to get some spectral results for the initial family of almost periodic equations.

We consider the Schr\"odinger operator $H$ on the half-line with a periodic potential $p$ plus a compactly supported potential $q$. For generic $p$, its essential spectrum has an infinite sequence of open gaps. We determine the asymptotics of the resonance counting function and show that, for sufficiently high energy, each non-degenerate gap contains exactly one eigenvalue or antibound state, giving asymptotics for their positions. Conversely, for any potential $q$ and for an...