Johanna Michor, Gerald Teschl
We investigate trace formulas for Jacobi 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 for the solutions of the Toda hierarchy in this class.
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Trace formulas for magnetic Schrödinger operators on periodic graphs and their applications
本文研究周期离散图上带周期磁势和电势的薛定谔算子，推导出用磁通量、电势和商图环集表示的迹公式，并据此获得总带宽的精确下界估计。
I prove that quasi-periodic Schrödinger operators in arbitrary dimension have some absolutely continuous spectrum.
We obtain the classical Ambarzumyan's theorem for the Sturm-Liouville operators $L_{t}(q)$ with $q\in L^{1}[0,1]$ and quasi-periodic boundary conditions, $t\in [0,2π)$, when there is not any additional condition on the potential $q$.
Finite and full scale localization for the multi-frequency quasi-periodic CMV matrices
研究多频准周期CMV矩阵的有限和全尺度局域化方法，提出与Goldstein等人关于多频准周期Schrödinger算子局域化结果相对应的CMV矩阵结果。
We prove that the size of the spectral gaps of weakly coupled quasi-periodic Schrödinger operators with Liouville frequencies decays exponentially. As an application, we obtain the homogeneity of the spectrum.
The spectrum of the multi-frequency quasi-periodic CMV matrices contains intervals
本文研究多频拟周期CMV矩阵的谱结构，在正Lyapunov指数和Diophantine条件下证明其谱在单位圆上包含区间。
We develop direct and inverse scattering theory for Jacobi operators which are short range perturbations of quasi-periodic finite-gap operators. We show existence of transformation operators, investigate their properties, derive the corresponding Gel'fand-Levitan-Marchenko equation, and find minimal scattering data which determine the perturbed operator uniquely.
Non-self-adjoint second-order differential pencils on a finite interval with non-separated quasi-periodic boundary conditions and jump conditions are studied. We establish properties of spectral characteristics and investigate the inverse spectral problem of recovering the operator from its spectral data. For this inverse problem we prove the corresponding uniqueness theorem and provide an algorithm for constructing its solution.
This article is concerned with the isospectral problem \[ -f'' + \frac{1}{4} f = zωf + z^2 \upsilon f \] for the periodic conservative Camassa-Holm flow, where $ω$ is a periodic real distribution in $H^{-1}_{\mathrm{loc}}(\mathbb{R})$ and $\upsilon$ is a periodic non-negative Borel measure on $\mathbb{R}$. We develop basic Floquet theory for this problem, derive trace formulas for the associated spectra and establish continuous dependence of these spectra on the coefficients with respect to a weak$^\ast$ topology.
We consider Schrödinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We determine trace formulas for the Schrödinger operators. The proof is based on the decomposition of the Schrödinger operators into a direct integral and a specific representation of fiber operators. The traces of the fiber operators are expressed as finite Fourier series of the quasimomentum. The coefficients of the Fourier series are given in terms of the potentials and cycles in the quotient graph from some specific cycle sets. We also present the tra…
Recently, a trace formula for non-self-adjoint periodic Schrödinger operators in $L^2(\mathbb{R})$ associated with Dirichlet eigenvalues was proved in [9]. Here we prove a corresponding trace formula associated with Neumann eigenvalues. In addition we investigate Dirichlet and Neumann eigenvalues of such operators. In particular, using the Dirichlet and Neumann trace formulas we provide detailed information on location of the Dirichlet and Neumann eigenvalues for the model operator with the potential $Ke^{2ix}$, where $K\in\mathbb{C}$.
We find a finite CMV matrix whose eigenvalues coincide with the Dirichlet data of a circular periodic problem. As a consequence, we obtain circular analogues of the classical trace formulae for periodic Jacobi matrices.
The present paper is devoted to studying the Hill-type formula and Krein-type trace formula for ODE, which is a continuous work of our previous work for Hamiltonian systems \cite{HOW}. Hill-type formula and Krein-type trace formula are given by Hill at 1877 and Krein in 1950's separately. Recently, we find that there is a closed relationship between them \cite{HOW}. In this paper, we will obtain the Hill-type formula for the $S$-periodic orbits of the first order ODEs. Such a kind of orbits is considered naturally to study the symmetric periodic and quasi-periodic solutions. By some similar id…
In this paper, we study block Jacobi operators on $\mathbb{Z}$ with quasi-periodic meromorphic potential. We prove the non-perturbative Anderson localization for such operators in the large coupling regime.
In this paper, we consider a family of Sturm-Liouville operators on the $ω$-periodic domain. The bifurcation with respect to the parameter region is studied, and the elliptic regions are estimated by trace formula. At last, these results are used to study the linear stability of the elliptic equilibrium point along $z$-axis in Robe's restricted three-body problem.
Construction of finite differentiable quasi-periodic Schrödinger operators with cantor spectrum
本文提出显式构造具有康托尔谱的 $C^k$ 拟周期薛定谔算子的方法，并给出谱间隙大小的多项式渐近估计。
We prove that at large disorder, with large probability and for a set of Diophantine frequencies of large measure, Anderson localization in $\Bbb Z^d$ is {\it stable} under localized time-quasi-periodic perturbations by proving that the associated quasi-energy operator has pure point spectrum. The main tools are the Fröhlich-Spencer mechanism for the random component and the Bourgain-Goldstein-Schlag mechanism for the quasi-periodic component. The formulation of this problem is motivated by questions of Anderson localization for non-linear Schrödinger equations.
In this paper, we study quasi-periodic CMV matrices with Verblunsky coefficients given by the skew-shift. We prove the positivity of Lyapunov exponents and Anderson localization for most frequencies, which establish the analogous results of one-dimensional Schrödinger operators proved by Bourgain, Goldstein and Schlag.
We consider operators of the form H+V where H is the one-dimensional harmonic oscillator and V is a zero-order pseudo-differential operator which is quasi-periodic in an appropriate sense (one can take V to be multiplication by a periodic function for example). It is shown that the eigenvalues of H+V have asymptotics of the form λ_n(H+V)=λ_n(H)+W(\sqrt n)n^{-1/4}+O(n^{-1/2}\ln(n)) as n\to+\infty, where W is a quasi-periodic function which can be defined explicitly in terms of V.
We establish Anderson localization for general analytic $k$-frequency quasi-periodic operators on $\mathbb{Z}^d$ for \textit{arbitrary} $k,d$.
We prove that the $1-d$ quantum harmonic oscillator is stable under spatially localized, time quasi-periodic perturbations on a set of Diophantine frequencies of positive measure. This proves a conjecture raised by Enss-Veselic in their 1983 paper \cite{EV} in the general quasi-periodic setting. The motivation of the present paper also comes from construction of quasi-periodic solutions for the corresponding nonlinear equation.
We discover that the distribution of (frequency and phase) resonances plays a role in determining the spectral type of supercritical quasi-periodic Schrödinger operators. In particular, we disprove the second spectral transition line conjecture of Jitomirskaya in the early 1990s.
We consider standard and extended CMV matrices with small quasi-periodic Verblunsky coefficients and show that on their essential spectrum, all spectral measures are purely absolutely continuous. This answers a question of Barry Simon from 2005.
Ballistic Transport for Schrödinger Operators with Quasi-periodic Potentials
该研究证明了任意维度d>1下，带一般准周期势的薛定谔算子存在弹道输运，推进了相关算子谱理论研究。
We consider CMV matrices, both standard and extended, with analytic quasi-periodic Verblunsky coefficients and prove Anderson localization in the regime of positive Lyapunov exponents. This establishes the CMV analog of a result Bourgain and Goldstein proved for discrete one-dimensional Schrödinger operators. We also prove a similar result for quantum walks on the integer lattice with suitable analytic quasi-periodic coins.
Anderson localization of long-range quasi-periodic operators via Dynamical Rigidity
本文通过动力学刚性论证，证明了长程准周期算子在大三角函数势和迪奥菲antine频率下的安德森局域化。
We consider the quasi-periodic Schrödinger operator with the non-degenerate Gevrey potential for the Diophantine frequency. We prove that if the coupling number of the potential is large, then the spectrum is homogeneous.
Quantum Dynamical Bounds for Quasi-Periodic Operators with Liouville Frequencies
针对具有刘维尔频率的拟周期薛定谔算子，结合克罗内克序列的半代数偏差估计与适配刘维尔情形的定量格林函数估计，建立了量子动力学上界。
We consider a one-frequency, quasi-periodic, block Jacobi operator, whose blocks are generic matrix-valued analytic functions. We establish Anderson localization for this type of operator under the assumption that the coupling constant is large enough but independent of the frequency. This generalizes a result of J. Bourgain and S. Jitomirskaya on localization for band lattice, quasi-periodic Schroedinger operators.
We consider a family of one frequency discrete analytic quasi-periodic Schrödinger operators which appear in [Bjer]. We show that this family provides an example of coexistence of absolutely continuous and point spectrum for some parameters as well as coexistence of absolutely continuous and singular continuous spectrum for some other parameters.
In this paper we study on $L^2(\mathbb{R}^d)$ the quasi-periodic Schrödinger operator $H=-Δ+ λV(x),$ where $V$ is a real analytic quasi-periodic function and $λ>0$. We first show that $H$ has no eigenvalues in \textit{low energy region}. We also provide in \textit{low energy region} the new phase transition parameter, i.e. the competition between the strength of coupling and the length for frequencies.
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