Related papers: Sharp Spectral Gaps, Arithmetic Localization, and …
We study ergodic Schr\"odinger operators defined over product dynamical systems in which one factor is periodic and the other factor is either a subshift over a finite alphabet or an irrational rotation of the circle. In the case in which…
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…
We generalize the approach to localization in one dimension introduced by Kunz-Souillard, and refined by Delyon-Kunz-Souillard and Simon, in the early 1980's in such a way that certain correlations are allowed. Several applications of this…
We construct six multi-parameter families of Hermitian quasi-exactly solvable matrix Schroedinger operators in one variable. The method for finding these operators relies heavily upon a special representation of the Lie algebra o(2,2) whose…
We introduce the periodic Airy-Schr\"odinger operator and we study its band spectrum. This is an example of an explicitly solvable model with a periodic potential which is not differentiable at its minima and maxima. We define a…
In $L_2 (\mathbb{R}^d; \mathbb{C}^n)$, we consider a selfadjoint matrix strongly elliptic second order differential operator $\mathcal{A}_\varepsilon$ with periodic coefficients depending on $\mathbf{x}/\varepsilon$. We find approximations…
We introduce computational strategies for measuring the ``size'' of the spectrum of bounded self-adjoint operators using various metrics such as the Lebesgue measure, fractal dimensions, the number of connected components (or gaps), and…
The Kotani-Last conjecture states that every ergodic operator in one space dimension with non-empty absolutely continuous spectrum must have almost periodic coefficients. This statement makes sense in a variety of settings; for example,…
Spectral equivalences of the quasi-exactly solvable sectors of two classes of Schrodinger operators are established, using Gaudin-type Bethe ansatz equations. In some instances the results can be extended leading to full isospectrality. In…
Sharp spectral asymptotics for 2- and 3-dimensional Schroedinger operators with strong magnetic field are derived under rather weak smoothness conditions. In comparison with version 1 of 2005 new results are added and minor errors…
We investigate trace formulas for one-dimensional Schroedinger operators which are trace class perturbations of quasi-periodic finite-gap operators using Krein's spectral shift theory. In particular, we establish the conserved quantities…
We consider semiclassical Schr\"odinger operators acting in $L^2(\mathbb{R}^d)$ with $d\geq3$. For these operators we establish a sharp spectral asymptotics without full regularity. For the counting function we assume the potential is…
We consider operators acting in $L^2(\mathbb{R}^d)$ with $d\geq3$ that locally behave as a magnetic Schr\"odinger operator. For the magnetic Schr\"odinger operators we suppose the magnetic potentials are smooth and the electric potential is…
In this paper, we prove that for any $d$-frequency analytic quasiperiodic Schr\"odinger operator, if the frequency is weak Liouvillean, and the potential is small enough, then the corresponding operator has absolutely continuous spectrum.…
Different cases of sequences of the Laplace Transformations for the 2D Schrodinger operator in the periodic magnetic field and electric potential are considered. They lead to the exactly solvable operators with nonstandard spectral…
We present a novel method for establishing large data local well-posedness in low regularity Sobolev spaces for general quasilinear Schr\"odinger equations with non-degenerate and nontrapping metrics. Our result represents a definitive…
We consider kernel operators defined by a dynamical system. The Hausdorff distance of spectra is estimated by the Hausdorff distance of subsystems. We prove that the spectrum map is $ \frac{1}{2} $-H\"older continuous provided the group…
We study discrete quasiperiodic Schr\"odinger operators on $\ell^2(\zee)$ with potentials defined by $\gamma$-H\"older functions. We prove a general statement that for $\gamma >1/2$ and under the condition of positive Lyapunov exponents,…
In this paper, we prove the generic version of Cantor spectrum for quasi-periodic Schr\"{o}dinger operators with finitely smooth and small potentials, and we also show pure point spectrum for a class of multi-frequency $C^k$ long-range…
We discuss Schr\"odinger operators on a half-line with decaying oscillatory potentials, such as products of an almost periodic function and a decaying function. We provide sufficient conditions for preservation of absolutely continuous…