Peter D. Hislop, Frederic Klopp, Jeffrey H. Schenker
We prove that the integrated density of states (IDS) associated to a random Schroedinger operator is locally uniformly Hoelder continuous as a function of the disorder parameter lambda. In particular, we obtain convergence of the IDS, as lambda tends to 0, to the IDS for the unperturbed operator at all energies for which the IDS for the unperturbed operator is continuous in energy.
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In this paper we discuss the continuity properties of the integrated density of states for random models based on that of the single site distribution. Our results are valid for models with independent randomness with arbitrary free parts. In particular in the case of the Anderson type models (with stationary, growing, decaying randomness) on the $ν$ dimensional lattice, with or without periodic and almost periodic backgrounds, we show that if the single site distribution is uniformly $α$-Hölder continuous, $ 0 < α\leq 1$, then the density of states is also uniformly $α$-Hölder continuous.
The integrated density of states of a Schroedinger operator with random potential given by a homogeneous Gaussian field whose covariance function is continuous, compactly supported and has positive mean, is locally uniformly Lipschitz-continuous. This is proven using a Wegner estimate.
It is proven that the kernel of the fermionic projector of regularized Dirac sea vacua in Minkowski Space is $L^4$-integrable. The proof is carried out in the specific setting of a continuous exponentially-decaying cutoff in momentum space. As a direct consequence, the corresponding causal Lagrangian is shown to be $L^1$-integrable. Some topological features of the integrated causal Lagrangian are analyzed. In particular, local Hölder-like estimates are proved for continuous regular variations of spacetime, of which a few examples are discussed. Particular emphasis is placed on first-order per…
We show Hölder continuity for the integrated density of states of a quasi-periodic Jacobi operator with analytic coefficients, in the regime of positive Lyapunov exponent and with a strong Diophantine condition on the frequency. In particular, when the coefficients are trigonometric polynomials we express the Hölder exponent in terms of the degrees of the coefficients.
Recently, Hislop and Marx studied the dependence of the integrated density of states on the underlying probability distribution for a class of discrete random Schrödinger operators, and established a quantitative form of continuity in weak* topology. We develop an alternative approach to the problem, based on Ky Fan inequalities, and establish a sharp version of the estimate of Hislop and Marx. We also consider a corresponding problem for continual random Schrödinger operators on $\R^d$.
The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schr{ö}dinger operator with magnetic field and a random potential which may be unbounded from above and below. In case that the magnetic field is constant and the random potential is ergodic and admits a so-called one-parameter decomposition, we prove the absolute continuity of the integrated density of states and provide explicit upper bounds on its derivative, the density of states. This local Lipschitz continuity of the integrated density…
Stability of the non-critical spectral properties I: arithmetic absolute continuity of the integrated density of states
本文证明了在特定算术频率条件下，非临界几乎 Mathieu 算子的频率无关解析扰动保持态密度绝对连续性。
The present paper is devoted to the study of spectral properties of random Schroedinger operators. Using a finite section method for Toeplitz matrices, we prove a Wegner estimate for some alloy type models where the single site potential is allowed to change sign. The results apply to the corresponding discrete model, too. In certain disorder regimes we are able to prove the Lipschitz continuity of the integrated density of states and/or localization near spectral edges.
Pointwise modulus of continuity of the Lyapunov exponent and integrated density of states for analytic multi-frequency quasiperiodic $M(2, \mathbb{C})$ cocycles
本文针对解析多频准周期余环，建立了 Lyapunov 指数在余环和频率上的逐点模量连续性，并由此得到态密度积分在能量、势和频率上的正则性。
We prove that the integrated density of states (IDS) of random Schrödinger operators with Anderson-type potentials on $L^2 (\R^d)$, for $d \geq1$, is locally Hölder continuous at all energies with the same Hölder exponent $0<α\leq1$ as the conditional probability measure for the single-site random variable. As a special case, we prove that if the probability distribution is absolutely continuous with respect to Lebesgue measure with a bounded density, then the IDS is Lipschitz continuous at all energies. The single-site potential $u\in L\_0^\infty (\R^d)$ must be nonnegative and compactly-supp…
Hölder continuity of the integrated density of states for Liouville frequencies
本文针对一般解析势能，在 Lyapunov 指数满足特定不等式条件下，证明了 Lyapunov 指数与态密度积分的 Hölder 连续性。
We first analyze the integrated density of states (IDS) of periodic Schrödinger operators on an amenable covering manifold. A criterion for the continuity of the IDS at a prescribed energy is given along with examples of operators with both continuous and discontinuous IDS'. Subsequently, alloy-type perturbations of the periodic operator are considered. The randomness may enter both via the potential and the metric. A Wegner estimate is proven which implies the continuity of the corresponding IDS. This gives an example of a discontinuous "periodic" IDS which is regularized by a random perturba…
We establish several properties of the integrated density of states for random quantum graphs: Under appropriate ergodicity and amenability assumptions, the integrated density of states can be defined using an exhaustion procedure by compact subgraphs. A trace per unit volume formula holds, similarly as in the Euclidean case. Our setting includes periodic graphs. For a model where the edge length are random and vary independently in a smooth way we prove a Wegner estimate and related regularity results for the integrated density of states. These results are illustrated for an example based on …
We show that the integrated density of states of the almost Mathieu operator is absolutely continuous if and only if the coupling is non-critical. We deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase, settling the measure-theoretical case of Problem 6 of Barry Simon's list of Schrödinger operator problems for the twenty-first century.
Anderson localization and Hölder continuity of the integrated density of states for analytic quasiperiodic Schrödinger operators
本文研究了准周期薛定谔算子在扰动范围内，对任意非常数解析势和任意迪ophantine频率的安德森局域化及积分态密度的霍尔德连续性，提出了一种新的格林函数控制方法以消除双重共振。
We study a class of continuous matrix-valued Anderson models acting on $L^{2}(\R^{d})\otimes \C^{N}$. We prove the existence of their Integrated Density of States for any $d\geq 1$ and $N\geq 1$. Then for $d=1$ and for arbitrary $N$, we prove the Hölder continuity of the Integrated Density of States under some assumption on the group $G_{μ_{E}}$ generated by the transfer matrices associated to our models. This regularity result is based upon the analoguous regularity of the Lyapounov exponents associated to our model, and a new Thouless formula which relates the sum of the positive Lyapounov e…
Hölder continuity of the integrated density of states for quasi-periodic Jacobi block matrices
本文证明了具Diophantine频率的离散拟周期d×d Jacobi块矩阵态密度积分的Hölder连续性，给出指数范围并推广且加强薛定谔算子情形结果，其证明基于新方案可从全局零计数得有限体积特征多项式局部零计数。
Hölder continuity, $|N_λ(E)-N_λ(E')|\le C |E-E'|^α$, with a constant $C$ independent of the disorder strength $λ$ is proved for the integrated density of states $N_λ(E)$ associated to a discrete random operator $H = H_o + λV$ consisting of a translation invariant hopping matrix $H_o$ and i.i.d. single site potentials $V$ with an absolutely continuous distribution, under a regularity assumption for the hopping term.
Using the fractional moment method it is shown that, within the Hartree-Fock approximation for the Disordered Hubbard Hamiltonian, weakly interacting Fermions at positive temperature exhibit localization, suitably defined as exponential decay of eigenfunction correlators. Our result holds in any dimension in the regime of large disorder and at any disorder in the one dimensional case. As a consequence of our methods, we are able to show Hölder continuity of the integrated density of states with respect to energy, disorder and interaction using known techniques.
We prove that the the density of states measure (DOSm) for random Schrödinger operators on $\mathbb{Z}^d$ is weak-$^*$ Hölder-continuous in the probability measure. The framework we develop is general enough to extend to a wide range of discrete, random operators, including the Anderson model on the Bethe lattice, as well as random Schrödinger operators on the strip. An immediate application of our main result provides quantitive continuity estimates for the disorder dependence of the DOSm and the integrated density of states (IDS) in the weak disorder regime. These results hold for a general …
Sharp Bounds for the Integrated Density of States of a Strongly Disordered 1D Anderson-Bernoulli Model
针对强无序下一维离散Anderson-Bernoulli模型的积分态密度，本文给出了全谱范围内与无序强度一致的上下界，证明存在不依赖无序参数的显式IDS取值的能量序列。
The regular tree Anderson model at low disorder
该研究证明了无限正则树安德森模型在低无序度下的离域性，填补谱缺失部分以扩展早期成果，论证可推广至特定无序情况，并证明无序度消失时李雅普诺夫指数的连续性。
It is shown that in a large class of disordered systems with non-degenerate disorder, in presence of non-local interactions, the Integrated Density of States (IDS) is at least Hölder continuous in one dimension and universally infinitely differentiable in higher dimensions. This result applies also to the IDS in any finite volume subject to the random potential induced by an ambient, infinitely extended disordered media. Dimension one is critical: in the Bernoulli case, within the class of exponential interactions, the IDS measure undergoes continuity phase transitions, from absolutely continu…
Phase transition in the Integrated Density of States of the Anderson model arising from a supersymmetric sigma model
本研究依托超对称σ模型已有结果，分析随机薛定谔算符的积分态密度渐近行为，证明三维及以上维度弱、强无序区域间的相变及强无序下无Lifshitz尾，还推导了Wegner型估计。
In this paper we establish an abstract, dynamical Thouless-type formula for affine families of $\mathrm{GL} (2,\mathbb{R})$ cocycles. This result extends the classical formula relating, via the Hilbert transform, the maximal Lyapunov exponent and the integrated density of states of a Schrödinger operator. Here, the role of the integrated density of states will be played by a more geometrical quantity, the fibered rotation number. As an application of this formula we present limitations on the modulus of continuity of random linear cocycles. Moreover, we derive Hölder-type continuity properties…
We study a matrix-valued Schrödinger operator with random point interactions. We prove the absence of absolutely continuous spectrum for this operator by proving that away from a discrete set its Lyapunov exponents do not vanish. For this we use a criterion by Gol'dsheid and Margulis and we prove the Zariski denseness, in the symplectic group, of the group generated by the transfer matrices. Then we prove estimates on the transfer matrices which lead to the Hölder continuity of the Lyapunov exponents. After proving the existence of the integrated density of states of the operator, we also prov…
We prove quantization of the Hall conductance for continuous ergodic Landau Hamiltonians under a condition on the decay of the Fermi projections. This condition and continuity of the integrated density of states are shown to imply continuity of the Hall conductance. In addition, we prove the existence of delocalization near each Landau level for these two-dimensional Hamiltonians. More precisely, we prove that for some ergodic Landau Hamiltonians there exists an energy $E$ near each Landau level where a ``localization length'' diverges. For the Anderson-Landau Hamiltonian we also obtain a tran…
We show there is a countable dense set of energies at which the integrated density of states of the 1D discrete Anderson-Bernoulli model can be given explicitly and does not depend on the disorder parameter, provided the latter is above an energy-dependent threshold.
We prove the Hölder continuity of the integrated density of states for a class of quasi-periodic long-range operators on $\ell^2(\Z^d)$ with large trigonometric polynomial potentials and Diophantine frequencies. Moreover, we give the Hölder exponent in terms of the cardinality of the level sets of the potentials, which improves, in the perturbative regime, the result obtained by Goldstein and Schlag \cite{gs2}. Our approach is a combination of Aubry duality, generalized Thouless formula and the regularity of the Lyapunov exponents of analytic quasi-periodic $GL(m,\C)$ cocycles which is proved …
We prove that the integrated density of surface states of continuous or discrete Anderson-type random Schroedinger operators is a measurable locally integrable function rather than a signed measure or a distribution. This generalize our recent results on the existence of the integrated density of surface states in the continuous case and those of A. Chahrour in the discrete case. The proof uses the new $L^p$-bound on the spectral shift function recently obtained by Combes, Hislop, and Nakamura. Also we provide a simple proof of their result on the Hoelder continuity of the integrated density o…
In this work we consider the Anderson model on $\ell^2(\mathbb{Z}^d)$ when the single site distribution (SSD) is given by $μ_1 * μ_2$, where $μ_1$ is the Cauchy distribution and $μ_2$ is any probability measure. For this model we prove that the integrated density of states (IDS) is infinitely differentiable irrespective of the disorder strength. Also, we investigate the local eigenvalue statistics of this model in $d\ge 2$, without any assumption on the localization property.
The subject of this work are random Schroedinger operators on regular rooted tree graphs $\T$ with stochastically homogeneous disorder. The operators are of the form $H_λ(ω) = T + U + λV(ω)$ acting in $\ell^2(\T)$, with $T $ the adjacency matrix, $U$ a radially periodic potential, and $V(ω)$ a random potential. This includes the only class of homogeneously random operators for which it was proven that the spectrum of $H_λ(ω)$ exhibits an absolutely continuous (ac) component; a results established by A. Klein for weak disorder, in case U=0 and $V(ω)$ given by iid random variables on $\T$. Our m…
Localization and Regularity of the Integrated Density of States for Schrödinger Operators on $\mathbb{Z}^d$ with $C^2$-cosine Like Quasi-periodic Potential
本文研究了具有 $C^2$-余弦类拟周期势的多维格点薛定谔算子，通过扩展多尺度分析方法，证明了 Anderson 局部化与积分态密度的 Hölder 连续性，并解决了高维情形下的特征值能级交叉问题。
We establish bounds on the density of states measure for Schrödinger operators. These are deterministic results that do not require the existence of the density of states measure, or, equivalently, of the integrated density of states. The results are stated in terms of a "density of states outer-measure" that always exists, and provides an upper bound for the density of states measure when it exists. We prove log-Hölder continuity for this density of states outer-measure in one, two, and three dimensions for Schrödinger operators, and in any dimension for discrete Schrödinger operators.
We derive bounds on the integrated density of states for a class of Schrödinger operators with a random potential. The potential depends on a sequence of random variables, not necessarily in a linear way. An example of such a random Schrödinger operator is the breather model, as introduced by Combes, Hislop and Mourre. For these models we show that the integrated density of states near the bottom of the spectrum behaves according to the so called Lifshitz asymptotics. This result can be used to prove Anderson localization in certain energy/disorder regimes.
We consider the Schroedinger equation with a supersymmetric random potential, where the superpotential is a Levy noise. We focus on the problem of computing the so-called complex Lyapunov exponent, whose real and imaginary parts are, respectively, the Lyapunov exponent and the integrated density of states of the system. In the case where the Levy process is non-decreasing, we show that the calculation of the complex Lyapunov exponent reduces to a Stieltjes moment problem, we ascertain the low-energy behaviour of the density of states in some generality, and relate it to the distributional prop…
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