Sergey A. Denisov
For a large class of Schrodinger operators, we introduce the hyperbolic quadratic pencils by making the coupling constant dependent on the energy in the very special way. For these pencils, many problems of scattering theory are easier to study. Then, we give some applications to the original Schrodinger operators.
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This paper deals with the $L_p$-spectrum of Schrödinger operators on the hyperbolic plane. We establish Lieb-Thirring type inequalities for discrete eigenvalues and study their dependence on $p$. Some bounds on individual eigenvalues are derived as well.
We show that for almost every frequency alpha \in \R \setminus \Q, for every C^omega potential v:\R/\Z \to R, and for almost every energy E the corresponding quasiperiodic Schrodinger cocycle is either reducible or nonuniformly hyperbolic. This result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic Schrodinger operator, and allows us to complete the proof of the Aubry-Andre conjecture on the measure of the spectrum of the Almost Mathieu Operator.
We study the asymptotic distribution of resonances for scattering by compactly supported potentials in hyperbolic space. We first establish an upper bound for the resonance counting function that depends only on the dimension and the support of the potential. We then establish the sharpness of this estimate by proving the a Weyl law for the resonance counting function holds in the case of radial potentials vanishing to some finite order at the edge of the support. As an application of the existence of potentials that saturate the upper bound, we derive additional resonance asymptotics that hol…
Schrödinger Operators with Potentials Generated by Hyperbolic Transformations: II. Large Deviations and Anderson Localization
本文研究由一致双曲变换轨道的赫尔德连续采样生成势的离散一维薛定谔算子，建立了大能量区域的一致大偏差估计，并证明了全谱安德森局域化。
We construct a class of matrix-valued Schrödinger operators with prescribed finite-band spectra of maximum spectral multiplicity. The corresponding matrix potentials are shown to be stationary solutions of the KdV hierarchy. The methods employed in this paper rely on matrix-valued Herglotz functions, Weyl--Titchmarsh theory, pencils of matrices, and basic inverse spectral theory for matrix-valued Schrödinger operators.
In this paper, a quadratic pencil of Schrödinger type difference operator $L_λ$ is taken under investigation to give a general perspective on the spectral analysis of non-selfadjoint difference equations of second order. Introducing Jost-type solutions, structural and quantitative properties of spectrum of the operator $L_λ$ are analyzed and hence, a discrete analog of the theory in Degasperis, (\emph{J.Math.Phys}. 11: 551--567, 1970) and Bairamov et. al, (\emph{Quaest. Math.} 26: 15--30, 2003) is developed. In addition, several analogies are established between difference and $q$-difference c…
We introduce concepts of essential numerical range for the linear operator pencil $λ\mapsto A-λB$. In contrast to the operator essential numerical range, the pencil essential numerical ranges are, in general, neither convex nor even connected. The new concepts allow us to describe the set of spectral pollution when approximating the operator pencil by projection and truncation methods. Moreover, by transforming the operator eigenvalue problem $Tx=λx$ into the pencil problem $BTx=λBx$ for suitable choices of $B$, we can obtain non-convex spectral enclosures for $T$ and, in the study of truncati…
We consider continuous $SL(2,\mathbb{R})$-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an $SO(2,\mathbb{R})$-cocycle. Using this, we show that if a cocycle's homotopy class does not display a certain obstruction to uniform hyperbolicity, then it can be $C^0$-perturbed to become uniformly hyperbolic. For cocycles arising from Schrödinger operators, the obstruction vanishes and we conclude that uniform hype…
We consider discrete one-dimensional Schrödinger operators whose potentials are generated by sampling along the orbits of a general hyperbolic transformation. Specifically, we show that if the sampling function is a non-constant Hölder continuous function defined on a subshift of finite type with an ergodic measure admitting a local product structure and a fixed point, then the Lyapunov exponent is positive away from a discrete set of energies. Moreover, for sampling functions in a residual subset of the space of Hölder continuous functions, the Lyapunov exponent is positive everywhere. If we …
Existence of resonances for Schrodinger operators on hyperbolic space
本文利用热迹与波迹展开及散射相的渐近性，证明了双曲空间上薛定谔算子在光滑紧支撑势下的共振存在性与下界。
We construct an explicit intertwining operator $\lcal$ between the Schrödinger group $e^{it \frac\Lap2} $ and the geodesic flow $g^t$ on certain Hilbert spaces of symbols on the cotangent bundle $T^* \X$ of a compact hyperbolic surface $\X = Γ\backslash \D$. Thus, the quantization Op(\lcal^{-1} a) satisfies an exact Egorov theorem. The construction of $\lcal$ is based on a complete set of Patterson-Sullivan distributions.
We consider the question of whether the high-energy eigenfunctions of certain Schrödinger operators on the $d$-dimensional hyperbolic space of constant curvature $-κ^2$ are flexible enough to approximate an arbitrary solution of the Helmholtz equation $Δh+h=0$ on $\mathbf{R}^d$, over the natural length scale $O(λ^{-1/2})$ determined by the eigenvalue $λ\gg 1$. This problem is motivated by the fact that, by the asymptotics of the local Weyl law, approximate Laplace eigenfunctions do have this approximation property on any compact Riemannian manifold. In this paper we are specifically interested…
The distribution of negative eigenvalues of Schrödinger operators on asymptotically hyperbolic manifolds
本文研究渐近双曲流形上薛定谔算子负特征值的计数函数渐近性，给出势条件判定有限或无限个负特征值，并在无限情形下得到区间$(-\infty,-E)$内特征值计数当$E\to 0$时的渐近行为。
We show that for a class of $C^2$ quasiperiodic potentials and for any Diophantine frequency, the spectrum of the corresponding Schrödinger operators is Cantor. Our approach is of purely dynamical systems, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to general $\mathrm{SL}(2,\mathbb R)$ cocycles, and obtain that uniform hyperbolic systems form a open and dense set in some one-parameter family.
For a large class of semiclassical pseudodifferential operators, including Schrödinger operators, $ P (h) = -h^2 Δ_g + V (x) $, on compact Riemannian manifolds, we give logarithmic lower bounds on the mass of eigenfunctions outside neighbourhoods of generic closed hyperbolic orbits. More precisely we show that if $ A $ is a pseudodifferential operator which is microlocally equal to the identity near the hyperbolic orbit and microlocally zero away from the orbit, then \[ \| u \| \leq C (\sqrt{\log(1/h)}/ h) \| P (h)u \| + C \sqrt {\log(1/h)} \| (I - A) u \| . \] This generalizes earlier estimat…
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