Lifschitz tails for the Anderson model

We present, in an expository way, an elementary rigorous proof (patterned after an argument of Kirsch-Martinelli) that the Anderson model has Lifschitz tails in very great generality.Research partially supported by USNSF Grant No. MCS-81-20833.     

On the Lifschitz singularity and the tailing in the density of states for random lattice systems

We prove rigorously the existence of a Lifschitz singularity in the density of states at zero energy in some random lattice systems of noninteracting bosons and fermions in any numberv of dimensions. The basic tool is a simple modification of the method of Fukushima to yield the correct upper and lower bounds for allv. We also comment on the mathematical difference between the models treated and the system of phonons with mass disorder in the harmonic approximation, whose behavior is known to be of Debye form, not Lifschitz, at low temperatures.Supported by the Swiss National Science Foundation.On leave of absence from the Institute de Fisica, University of São Paulo, Brazil.     

Lifshitz tails for periodic plus random potentials

We prove that the integrated density of states () for a potentialW =V _per +V has Lifshitz tails where V_per is a periodic potential with reflection symmetry andV is a random potential, e.g., of the formV =q _i ()f(x–i).research partially supported by DFG.research partially supported by USNSF under grant No. MCS-81-20833.     

Lifschitz singularities for periodic operators plus random potentials

UsingX-bounding (lower bounds by Laplacians with mixed boundary conditions and discrete analogs), we obtain the Lifschitz exponent at the bottom of the spectrum for random operators of typeH =T+V , withT a (periodic) generator of a positivity-preserving semigroup, extending results by Kirsch and Simon.     

Regularity of the density of states in the Anderson model on a strip for potentials with singular continuous distributions

We derive regularity properties for the density of states in the Anderson model on a one-dimensional strip for potentials with singular continuous distributions. For example, if the characteristic function is infinitely differentiable with bounded derivatives and together with all its derivatives goes to zero at infinity, we show that the density of states is infinitely differentiable.     

Lifshits Tails Caused by Anisotropic Decay: The Emergence of a Quantum-Classical Regime

We investigate Lifshits-tail behaviour of the integrated density of states for a wide class of Schrödinger operators with positive random potentials. The setting includes alloy-type and Poissonian random potentials. The considered (single-site) impurity potentials f: ?^d→[0,∞[ decay at infinity in an anisotropic way, for example, \(f(x_{1},x_{2})\sim (|x_{1}|^{\alpha_{1}}+|x_{2}|^{\alpha_{2}})^{-1}\) as |(x₁,x₂)|→∞. As is expected from the isotropic situation, there is a so-called quantum regime with Lifshits exponent d/2 if both α₁ and α₂ are big enough, and there is a so-called classical regime with Lifshits exponent depending on α₁ and α₂ if both are small. In addition to this we find two new regimes where the Lifshits exponent exhibits a mixture of quantum and classical behaviour. Moreover, the transition lines between these regimes depend in a nontrivial way on α₁ and α₂ simultaneously.     

Lifshitz Asymptotics via Linear Coupling of Disorder

We present a simple method for proving Lifshitz asymptotics for random Schrödinger operators and apply it to the Anderson and Poisson model.     

The Band-Edge Behavior of the Density of Surfacic States

This paper is devoted to the asymptotics of the density of surfacic states near the spectral edges for a discrete surfacic Anderson model. Two types of spectral edges have to be considered: fluctuating edges and stable edges. Each type has its own type of asymptotics. In the case of fluctuating edges, one obtains Lifshitz tails the parameters of which are given by the initial operator suitably 'reduced' to the surface. For stable edges, the surface density of states behaves like the surface density of states of a constant (equal to the expectation of the random potential) surface potential. Among the tools used to establish this are the asymptotics of the surface density of states for constant surface potentials.Mathematics Subject Classifications (2000) 35P20, 46N50, 47B80.     

Analyticity of the density of states in the anderson model on the Bethe lattice

Let H=1/2+V on l²(B), whereB is the Bethe lattice andV(x),x B, are i.i.d.r.v.'s with common probability distribution. It is shown that for distributions sufficiently close to the Cauchy distribution, the density of states(E) is analytic in a strip about the real axis.     

Long time tails in stationary random media II: Applications

In a previous paper we developed a mode-coupling theory to describe the long time properties of diffusion in stationary, statistically homogeneous, random media. Here the general theory is applied to deterministic and stochastic Lorentz models and several hopping models. The mode-coupling theory predicts that the amplitudes of the long time tails for these systems are determined by spatial fluctuations in a coarse-grained diffusion coefficient and a coarse-grained free volume. For one-dimensional models these amplitudes can be evaluated, and the mode-coupling theory is shown to agree with exact solutions obtained for these models. For higher-dimensional Lorentz models the formal theory yields expressions which are difficult to evaluate. For these models we develop an approximation scheme based upon projecting fluctuations in the diffusion coefficient and free volume onto fluctuations in the density of scatterers.Work supported by grant No. CHE 77-16308 from the National Science Foundation and by a Nato Travel Grant.     

Lifshitz Tails for Acoustic Waves in Random Quantum Waveguide

In this study, we consider acoustic operators in a random quantum waveguide. Precisely we deal with an elliptic operator in the divergence form on a random strip. We prove that the integrated density of states of the relevant operator exhibits Lifshitz behavior at the bottom of the spectrum. This result could be used to prove localization of acoustic waves at the bottom of the spectrum. 2000 Mathematics Subject Classification: 81Q10, 35P05, 37A30, 47F05     

The density of states in the Anderson model at weak disorder: A renormalization group analysis of the hierarchical model

We study the density of states in a hierarchical approximation of the Anderson tight-binding model at weak disorder using a renormalization group approach. Since the Laplacian term in our model is hierarchical, the renormalization group transformations act essentially on the local potential distribution and the energy. Technically, we use the supersymmetric replica trick and study the averaged Green's function. Starting with a Gaussian distribution with small variance, we find that the density of states is analytic as soon as the variance of the potential is turned on, except possibly near the band edge, where we can show this only for>2, which corresponds tod>4. Moreover, it is perturbatively close to the free one, except near the eigenvalues of the (hierarchical) Laplacian, where it is given (up to perturbative corrections) by the rescaled potential distribution.     

Band Edge Behavior of the Integrated Density of States of Random Jacobi Matrices in Dimension 1

Let H be a Jacobi matrix acting on and V a random potential of Anderson type. Let H = H+V . We give a general formula relating the decay of the integrated density of states of H at the edges of the almost sure spectrum of H to the decay of the integrated density of states of H at the edges of the spectrum of H.     

Asymptotics of the Interband Light Absorption Coefficient near the Band Edge for an Alloy-Type Model

We find the asymptotics of the interband light absorption coefficient of an alloy-type model in the case when the ground-state energies of the electron and the hole Hamiltonians are finite.     

Weak disorder expansion of the invariant measure for the one-dimensional Anderson model

We show that the formal perturbation expansion of the invariant measure for the Anderson model in one dimension has singularities at all energiesE ₀=2 cos(p/q); we derive a modified expansion near these energies that we show to have finite coefficients to all orders. Moreover, we show that the firstq–3 of them coincide with those of the naive expansion, while there is an anomaly in the (q–2)th term. This also gives a weak disorder expansion for the Liapunov exponent and for the density of states. This generalizes previous results of Kappus and Wegner and of Derrida and Gardner.     

Non-Lifshitz Tails at the Spectral Bottom of Some Random Operators

In this paper we continue with the investigation of the behavior of the integrated density of states of random operators of the form H _ω =−∇ ρ _ω ∇. In the present work we are interested in its behavior at 0, the bottom of the spectrum of H _ω . We prove that it converges exponentially fast to the integrated density of states of some periodic operator . Being periodic, cannot exhibit a Lifshitz behaviour. This result relates to the result of S.M. Kozlov (Russ. Math. Surv. 34(4):168–169, 1979) and improves it. Research partially supported by the Research Unity 01/UR/ 15-01 projects.     

Lifshitz tails and long-time decay in random systems with arbitrary disorder

In random systems, the density of states of various linear problems, such as phonons, tight-binding electrons, or diffusion in a medium with traps, exhibits an exponentially small Liftshitz tail at band edges. When the distribution of the appropriate random variables (atomic masses, site energies, trap depths) has a delta function at its lower (upper) bound, the Lifshitz singularities are pure exponentials. We study in a quantitative way how these singularities are affected by a universal logarithmic correction for continuous distributions starting with a power law. We derive an asymptotic expansion of the Lifshitz tail to all orders in this logarithmic variable. For distributions starting with an essential singularity, the exponent of the Lifshitz singularity itself is modified. These results are obtained in the example of harmonic chains with random masses. It is argued that analogous results hoid in higher dimensions. Their implications for other models, such as the long-time decay in trapping problems, are also discussed.     

Random Operators and Crossed Products

This article is concerned with crossed products and their applications to random operators. We study the von Neumann algebra of a dynamical system using the underlying Hilbert algebra structure. This gives a particularly easy way to introduce a trace on this von Neumann algebra. We review several formulas for this trace, show how it comes as an application of Connes" noncommutative integration theory and discuss Shubin"s trace formula. We then restrict ourselves to the case of an action of a group on a group and include new proofs for some theorems of Bellissard and Testard on an analogue of the classical Plancherel theorem. We show that the integrated density of states is a spectral measure in the periodic case, thereby generalizing a result of Kaminker and Xia. Finally, we discuss duality results and apply a method of Gordon et al. to establish a duality result for crossed products by Z.     

Long time tails in stationary random media. I. Theory

Diffusion of moving particles in stationary disordered media is studied using a phenomenological mode-coupling theory. The presence of disorder leads to a generalized diffusion equation, with memory kernels having power law long time tails. The velocity autocorrelation function is found to decay like t^–(d/2+1), while the time correlation function associated with the super-Burnett coefficient decays liket ^–d/2 for long times. The theory is applicable to a wide variety of dynamical and stochastic systems including the Lorentz gas and hopping models. We find new, general expressions for the coefficients of the long time tails which agree with previous results for exactly solvable hopping models and with the low-density results obtained for the Lorentz gas. Finally we mention that if the moving particles are charged, then the long time tails imply that there is an ^d/2 contribution to the low-frequency part of the frequency-dependent electrical conductivity.