S. A. Denisov
We prove that the massless Dirac operator in $\mathbb{R^3}$ with long-range potential has an a.c. spectrum which fills the whole real line. The Dirac operators with matrix-valued potentials are considered as well.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We prove that 3-dimensional Schrodinger operator with slowly decaying potential has an absolutely continuous spectrum that fills the positive half-line. The asymptotics of Green's function is obtained as well.
We give a new proof of a version of Klein's theorem on the existence of absolutely continuous spectrum for the Anderson model on the Bethe Lattice at weak disorder.
The justification of the absolutely continuous spectrum eigenfunctions asymptotics in the problem of three one-Dimensional short-range quantum particles with repulsion has been done.
We prove persistence of absolutely continuous spectrum for the Anderson model on a general class of tree-like graphs.
We consider a polyharmonic operator $H=(-Δ)^l+V(x)$ in dimension two with $l\geq 6$, $l$ being an integer, and a limit-periodic potential $V(x)$. We prove that the spectrum contains a semiaxis of absolutely continuous spectrum.
We prove the almost sure existence of absolutely continuous spectrum at low disorder for the Anderson model on the simplest example of a product of a regular tree with a finite graph. This graph contains loops of unbounded size.
Neat stuff about eigenfunctions, transfer matrices, and a.c. spectrum of one-dimensional Schrodinger operators
We discuss the dynamical implications of the recent proof that for a quantum particle in a random potential on a regular tree graph absolutely continuous spectrum occurs non-perturbatively through rare fluctuation-enabled resonances. The main result is spelled in the title.
For a bounded real-valued function $V$ on ${\Bbb R}^d$, we consider two Schrödinger operators $H_+=-Δ+V$ and $H_-=-Δ-V$. We prove that if the negative spectra $H_+$ and $H_-$ are discrete and the negative eigenvalues of $H_+$ and $H_-$ tend to zero sufficiently fast, then the absolutely continuous spectra cover the positive half-line $[0,\infty)$.
It is proven that the absolutely continuous spectrum of matrix Schrödinger operators coincides (with the multiplicity taken into account) with the spectrum of the unperturbed operator if the (matrix) potential is square integrable. The same result is also proven for some classes of slower decaying potentials if they are smooth.
We provide a simplified version of the geometric method given by Froese, Hasler and Spitzer and use it to prove the existence of absolutely continuous spectrum for a Cayley tree of arbitrary degree k.
We give sufficient conditions for the presence of the absolutely continuous spectrum of a Schrödinger operator on a regular rooted tree without loops (also called regular Bethe lattice or Cayley tree).
Quasi-one-dimensional stochastic Dirac operators with an odd number of channels, time reversal symmetry but otherwise efficiently coupled randomness are shown to have one conducting channel and absolutely continuous spectrum of multiplicity two. This follows by adapting the criteria of Guivarch-Raugi and Goldsheid-Margulis to the analysis of random products of matrices in the group SO$^*(2L)$, and then a version of Kotani theory for these operators. Absence of singular spectrum can be shown by adapting an argument of Jaksic-Last if the potential contains random Dirac peaks with absolutely cont…
We prove an averaging formula for the derivative of the absolutely continuous part of the density of states measure for an ergodic family of CMV matrices. As a consequence, we show that the spectral type of such a family is almost surely purely absolutely continuous if and only if the density of states is absolutely continuous and the Lyapunov exponent vanishes almost everywhere with respect to the same. Both of these results are CMV operator analogues of theorems obtained by Kotani for Schrödinger operators.
We consider a family of operators $-Δ+ t V$ with a slowly decaying and oscillating potential $V$. We prove that the absolutely continuous spectrum of this operator is essentially supported by $[0,\infty)$ for almost every $t$.
We give an example of an operator with different weak and strong absolutely continuous subspaces, and a counterexample to the duality problem for the spectral components. Both examples are optimal in the scale of compact operators.
The purpose of this paper is to prove that the spectrum of an isotropic Maxwell operator with electric permittivity and magnetic permeability that are periodic along certain directions and tending to a constant super-exponentially fast in the remaining directions is purely absolutely continuous. The basic technical tools is a new ``operatorial'' identity relating the Maxwell operator to a vector-valued Schrodinger operator. The analysis of the spectrum of that operator is then handled using ideas developed by the same authors in a previous paper.
Modifying Rudin's original construction of the Rudin-Shapiro sequence, we derive a new substitution-based sequence with purely absolutely continuous diffraction spectrum.
In this paper we consider two classes of random Hamiltonians on $L^2(\RR^d)$ one that imitates the lattice case and the other a Schrödinger operator with non-decaying, non-sparse potential both of which exhibit a.c. spectrum. In the former case we also know the existence of dense pure point spectrum for some disorder thus exhibiting spectral transition valid for the Bethe lattice and expected for the Anderson model in higher dimension.
We formulate and prove the existence and uniqueness of the generalized Fourier transform associated with the absolutely continuous part of an arbitrary selfadjoint operator on a separable Hilbert space. To this end we develop a novel method to decompose an absolutely continuous operator into a variable fiber direct integral of selfadjoint operators.
We consider a family of random Schrödinger operators on the discrete strip with decaying random $\ell^2$ matrix potential. We prove that the spectrum is almost surely pure absolutely continuous, apart from random, possibly embedded eigenvalues, which may accumulate at band edges.
We characterize the absolutely continuous spectrum of half-line one-dimensional Schrödinger operators in terms of the limiting behavior of the Crystaline Landauer-Büttiker conductance of the associated finite samples.
By generalising Rudin's construction of an aperiodic sequence, we derive new substitution-based structures which have purely absolutely continuous diffraction and mixed dynamical spectrum, with absolutely continuous and pure point parts. We discuss several examples, including a construction based on Fourier matrices which yields constant-length substitutions for any length.
I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of one-dimensional Schr"odinger operators. These include an Oracle Theorem that predicts the potential and rather general results on the approach to certain limit potentials. In particular, we prove a Denisov-Rakhmanov type theorem for the general finite gap case. The main theme is the following: It is extremely difficult to produce absolutely continuous spectrum in one space dimension and thus its existence has strong implications.
We prove new criteria of stability of the absolutely continuous spectrum of one-dimensional Schrödinger operators under slowly decaying perturbations. As applications, we show that the absolutely continuous spectrum of the free and periodic Schrödinger operators is preserved under perturbations by all potentials $V(x)$ satisfying $|V(x)| \leq C(1+x)^{-\frac{2}{3}-ε}.$ The main new technique includes an a.e.\ convergence theorem for a class of integral operators.
We disprove a conjecture of Breuer-Last-Simon concerning the absolutely continuous spectrum of Jacobi matrices with coefficients that obey an $\ell^2$ bounded variation condition with step $q$. We prove existence of a.c. spectrum on a smaller set than that specified by the conjecture and prove that our result is optimal.
We prove that the spectrum of the almost Mathieu operator is absolutely continuous if and only if the coupling is subcritical. This settles Problem 6 of Barry Simon's list of Schrödinger operator problems for the twenty-first century.
We investigate absolutely continuous spectrum of generalized indefinite strings. By following an approach of Deift and Killip, we establish stability of the absolutely continuous spectra of two model examples of generalized indefinite strings under rather wide perturbations. In particular, one of these results allows us to prove that the absolutely continuous spectrum of the isospectral problem associated with the conservative Camassa-Holm flow in the dispersive regime is essentially supported on the interval $[1/4,\infty)$.
We study the spectrum of spherically symmetric Dirac operators in three-dimensional space with potentials tending to infinity at infinity under weak regularity assumptions. We prove that purely absolutely continuous spectrum covers the whole real line if the potential dominates the mass, or scalar potential, term. In the situation where the potential and the scalar potential are identical, the positive part of the spectrum is purely discrete; we show that the negative half-line is filled with purely absolutely continuous spectrum in this case.
We show that a generic quasi-periodic Schrödinger operator in $L^2(\mathbb{R})$ has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that for each of these sampling functions, the Schrödinger operator with the resulting potential has empty absolutely continuous spectrum.
We present general principles for the preservation of a.c. spectrum under weak perturbations. The Schrodinger operators on the strip and on the Caley tree (Bethe lattice) are considered.
We consider multi-type Galton Watson trees that are close to a tree of finite cone type in distribution. Moreover, we impose that each vertex has at least one forward neighbor. Then, we show that the spectrum of the Laplace operator exhibits almost surely a purely absolutely continuous component which is included in the absolutely continuous spectrum of the tree of finite cone type.
Absolutely continuous edge spectrum of topological insulators with an odd time-reversal symmetry
该研究证明受奇时间反演对称性保护的非平凡二维拓扑绝缘体具有绝对连续边谱，借助Wold分解的时间反演对称版本分离出其弹道边缘模，为相关拓扑绝缘体的边缘态性质提供了理论证明。
We consider standard subordinacy theory for general Sturm--Liouville operators and give criteria when boundedness of solutions implies that no subordinate solutions exist. As applications, we prove a Weidmann-type result for general Sturm--Liouville operators and investigate the absolutely continuous spectrum of radially symmetric quantum trees.
We continue to investigate absolutely continuous spectrum of generalized indefinite strings. By following an approach of Deift and Killip, we establish stability of the absolutely continuous spectra of two more model examples of generalized indefinite strings under rather wide perturbations. In particular, one of these results allows us to prove that the absolutely continuous spectrum of the isospectral problem associated with the two-component Camassa-Holm system in a certain dispersive regime is essentially supported on the set $(-\infty,-1/2]\cup [1/2,\infty)$.
Absolutely continuous spectrum for truncated topological insulators
本文证明沿足够大分隔曲线截断的拓扑绝缘体边缘系统具有绝对连续谱，方法结合了分离几何与内禀电导的体边对应关系及直边绝对连续谱结果。
We consider a family of multi-dimensional Schrödinger operators $-Δ+t V$ with a real $t$. The potential $V$ in our model decays at infinity in a special way, so that it satisfies a certain integral condition. We prove that the absolutely continuous spectrum of this operator covers the interval $[0,\infty)$ for almost every $t$.
This paper deals with general structural properties of one-dimensional Schr"odinger operators with some absolutely continuous spectrum. The basic result says that the omega limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure. This implies an Oracle Theorem for such potentials and Denisov-Rakhmanov type theorems. In the discrete case, for Jacobi operators, these issues were discussed in my recent paper [19]. The treatment of the continuous case in the present paper depends on the same basic ideas.
On an infinite, radial metric tree graph we consider the corresponding Laplacian equipped with self-adjoint vertex conditions from a large class including $δ$- and weighted $δ'$-couplings. Assuming the numbers of different edge lengths, branching numbers and different coupling conditions to be finite, we prove that the presence of absolutely continuous spectrum implies that the sequence of geometric data of the tree as well as the coupling conditions are eventually periodic. On the other hand, we provide examples of self-adjoint, non-periodic couplings which admit absolutely continuous spectru…
Recent results of Denisov and Kaluzhny-Shamis describe the absolutely continuous spectrum of Jacobi matrices with coefficients that obey an l^2 bounded variation condition with step p and are asymptotically periodic. We extend these results to orthogonal polynomials on the unit circle. We also replace the asymptotic periodicity condition by the weaker condition of convergence to an isospectral torus and, for p=1 and p=2, we remove even that condition.
We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely continuous spectrum. Then we establish stability of the absolutely continuous spectrum in more general situations, where imbedded singular spectrum may occur. We present two kinds of optimal conditions for the stability of absolutely continuous spectrum: decay and smoothness. In the decay direction, we show that a sufficient (in the power scale) condition is $|…
The presence of chiral modes on the edges of quantum Hall samples is essential to our understanding of the quantum Hall effect. In particular, these edge modes should support ballistic transport and therefore, in a single particle picture, be supported in the absolutely continuous spectrum of the single-particle Hamiltonian. We show in this note that if a free fermion system on the two-dimensional lattice is gapped in the bulk, and has a nonvanishing Hall conductance, then the same system put on a half-space geometry supports edge modes whose spectrum fills the entire bulk gap and is absolutel…
We apply the methods of classical approximation theory (extreme properties of polynomials) to study the essential support $Σ_{ac}$ of the absolutely continuous spectrum of Jacobi matrices. First, we prove an upper bound on the measure of $Σ_{ac}$ which takes into account the value distribution of the diagonal elements, and implies the bound due to Deift-Simon and Poltoratski-Remling. Second, we generalise the differential inequality of Deift-Simon for the integrated density of states associated with the absolutely continuous spectrum to general Jacobi matrices.
We study the spectrum of random operators on a large class of trees. These trees have finitely many cone types and they can be constructed by a substitution rule. The random operators are perturbations of Laplace type operators either by random potentials or by random hopping terms, i.e., perturbations of the off-diagonal elements. We prove stability of arbitrary large parts of the absolutely continuous spectrum for sufficiently small but extensive disorder.
A scattering zipper is a system obtained by concatenation of scattering events with equal even number of incoming and out going channels. The associated scattering zipper operator is the unitary equivalent of Jacobi matrices with matrix entries. For infinite identical events and random phases, Lyapunov exponents positivity is proved and yields to the absence of absolutely continuous spectrum.
We will consider cross products of finite graphs with a class of trees that have arbitrarily but finitely long line segments, such as the Fibonacci tree. Such cross products are called tree-strips. We prove that for small disorder random Schrödinger operators on such tree-strips have purely absolutely continuous spectrum in a certain set.
The absolutely continuous spectrum of one-dimensional Schrödinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectrum of free and periodic Schrödinger operators is preserved under all perturbations $V(x)$ satisfying $|V(x)|\leq C(1+x)^{-α}$, $α>\frac{1}{2}.$ This result is optimal in the power scale. More general classes of perturbing potentials which are not necessarily power decaying are also treated. A general criterion for stability of the absolutely continuous spectrum of one-dimensional Schr…
A tree-strip of finite cone type is the product of a tree of finite cone type with a finite set. We consider random Schrödinger operators on these tree strips, similar to the Anderson model. We prove that for small disorder the spectrum is almost surely, purely, absolutely continuous in a certain set.
Quasiperiodic tilings are often considered as structure models of quasicrystals. In this context, it is important to study the nature of the diffraction measures for tilings. In this article, we investigate the diffraction measures for S-adic tilings in R^d, which are constructed from a family of geometric substitution rules. In particular, we firstly give a sufficient condition for the absolutely continuous component of the diffraction measure for an S-adic tiling to be zero. Next, we prove this sufficient condition for "almost all" binary block-substitution cases and thus prove the absence o…
We study the spectra of quantum trees of finite cone type. These are quantum graphs whose geometry has a certain homogeneity, and which carry a finite set of edge lengths, coupling constants and potentials on the edges. We show the spectrum consists of bands of purely absolutely continuous spectrum, along with a discrete set of eigenvalues. Afterwards, we study random perturbations of such trees, at the level of edge length and coupling, and prove the stability of pure AC spectrum, along with resolvent estimates.
We review a geometric approach to proving absolutely continuous (ac) spectrum for random and deterministic Schrödinger operators developed in \cite{FHS1,FHS2,FHS3,FHS4}. We study decaying potentials in one dimension and present a simplified proof of ac spectrum of the Anderson model on trees. The latter implies ac spectrum for a percolation model on trees. Finally, we introduce certain loop tree models which lead to some interesting open problems.
On absolutely continuous spectrum for one-channel unitary operators
本文通过建立径向转移矩阵形式化与Carmona公式的类似物，推导绝对连续谱判据，并应用于周期散射拉链的随机Hilbert-Schmidt扰动。
In the model suggested by Smilansky one studies an operator describing the interaction between a quantum graph and a system of $K$ one-dimensional oscillators attached at several different points in the graph. The present paper is the first one in which the case $K>1$ is investigated. For the sake of simplicity we consider K=2, but our argument is of a general character. In this first of two papers on the problem, we describe the absolutely continuous spectrum. Our approach is based upon scattering theory.
We prove for the square Fibonacci Hamiltonian that the density of states measure is absolutely continuous for almost all pairs of small coupling constants. This is obtained from a new result we establish about the absolute continuity of convolutions of measures arising in hyperbolic dynamics with exact-dimensional measures.
We consider random Schrödinger operators on tree graphs and prove absolutely continuous spectrum at small disorder for two models. The first model is the usual binary tree with certain strongly correlated random potentials. These potentials are of interest since for complete correlation they exhibit localization at all disorders. In the second model we change the tree graph by adding all possible edges to the graph inside each sphere, with weights proportional to the number of points in the sphere.
Absence of Absolutely Continuous Spectrum for One-Dimensional Capacitance Operators with Critical Long-Range Interactions
本文证明了一维临界长程相互作用电容算子绝对连续谱的缺失，通过有限差分界绕过四次衰减要求，为波系统Anderson局域化提供关键步骤。
We consider the discrete Schrödinger operator $H=-Δ+V$ with a sparse potential $V$ and find conditions guaranteeing either existence of wave operators for the pair $H$ and $H_0=-Δ$, or presence of dense purely point spectrum of the operator $H$ on some interval $[λ_0,0]$ with $λ_0<0$.
Disorder on the hyperbolic square lattice I: Anderson delocalization and absolutely continuous spectrum
研究双曲格上的安德森模型，证明弱无序下正测度集合上绝对连续谱存在且无奇异谱，并在特定条件下获得纯绝对连续谱。
Bethe-Sommerfeld Conjecture and Absolutely Continuous Spectrum of Multi-Dimensional Quasi-Periodic Schrödinger Operators
本文研究了多维准周期Schrödinger算子的绝对连续谱，证明了一般算子的谱包含半轴，构造了本征函数，并基于动量空间多尺度分析完成证明。
Notes on SU(2) invariant absolutely continuous probability measures
本文利用紧Hausdorff拓扑群上Haar测度的存在性，证明了$SU(2)$不变连续概率联合分布下$\mathbb{C}^2$的Lebesgue可测凸锥的测度独立性性质。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。