arXiv subjects
Publications and source records attributed to Victor Chulaevsky.
This paper is a follow-up of our earlier work [11] where a uniform exponential Anderson localization was proved for a class of deterministic (including quasi-periodic) Hamiltonians with the help of a variant of the KAM (Kolmogorov--Arnold--Moser) approach. Building on [11], we prove for the same class of operators a non-local variant of the Minami eigenvalue concentration estimate.
An ensemble of quasi-periodic discrete Schrödinger operators with an arbitrary number of basic frequencies is considered, in a lattice of arbitrary dimension, in which the hull function is a realisation of a stationary Gaussian process on the torus. We show that, for almost every element of the ensemble, the quasi-periodic operator boasts Anderson localization with simple pure point spectrum at strong coupling. One of the ingredients of the proof is a new lower bound on the interpolation error for stationary Gaussian processes on the torus (also known as local non-determinism).
Following [7,8], we analyze regularity properties of single-site probability distributions of the random potential and of the Integrated Density of States (IDS) in the Anderson models with infinite-range interactions and arbitrary nontrivial probability distributions of the site potentials. In the present work, we study $2$-particle Anderson Hamiltonians on a lattice and prove spectral and strong dynamical localization at low energies, with exponentially decaying eigenfunctions, for a class of site potentials featuring a power-law decay.
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 continuous to singular continuous behaviour (the singularity in the latter case was known before). The continuity transitions do not occur for sub-exponential or slower decaying interactions, nor for $d\ge 2$. Technically, the case of polynomial decay is the simplest one. The proposed approach provides a complement to the classical Wegner estimate which says, essentially, that the IDS in the short-range models is at least as regular as the marginal distribution of the disorder. In the models with non-local interaction the IDS is actually much more regular than the underlying disorder, which can even be discrete, due to the smoothing effect of multiple convolutions. In turn, smoothness of the IDS is responsable for a mechanism complementing the usual Lifshitz tails phenomenon. It is also shown that the disorder can take various forms (e.g., substitution or random displacements) and need not be stochastically stationary (as in Delone--Anderson or trim\-med/crooked Hamiltonians, for example); this does not affect the main phenomena observed already in the simplest setting. Long-range models have an amazingly large number of connections to several classical problems of harmonic analysis, probability theory, dynamical systems and number theory.
Following [5], we analyze regularity properties of single-site probability distributions of the random potential and of the Integrated Density of States (IDS) in the Anderson models with infinite-range interactions. In the present work, we study in detail a class of polynomially decaying interaction potentials of rather artificial (piecewise-constant) form, and give a complete proof of infinite smoothness of the IDS in an arbitrarily large finite domain subject to the fluctuations of the entire, infinite random environment. A variant of this result, based as in [5] on the harmonic analysis of probability measures, results in a proof of spectral and dynamical Anderson localization in the considered models.
We study the 1d Schrödinger operators with alloy type random supercritical decaying potential and prove the clock convergence for the local statistics of eigenvalues. We also consider, besides the standard i.i.d. case, more general ones with exponentially decaying correlations.
We propose a reformulation of the bootstrap version of the Multi-Scale Analysis (BMSA), developed by Germinet and Klein, to make explicit the fact that BMSA implies asymptotically exponential decay of eigenfunctions (EFs) and of EF correlators (EFCs), in the lattice Anderson models with diagonal disorder, viz. with an IID random potential. We also show that the exponential scaling limit of EFs and EFCs holds true for a class of marginal distributions of the random potential with regularity lower than Hölder continuity of any positive order.
We give an improved estimate for the regularity of the conditional distribution of the empiric mean of a finite sample of IID random variables, conditional on the sample "fluctuations", extending the well-known property of Gaussian IID samples. Specifically, we replace the bounds in probability, established in our earlier works, by those in distribution, and this results in the optimal regularity exponent in the final estimate.
We discuss the techniques and results of the multi-particle Anderson localization theory for disordered quantum systems with nontrivial interaction. After a detailed presentation of the approach developed earlier by Aizenman and Warzel, we extend their results to the models with exponentially decaying, infinite-range interaction.
This short note is a complement to our recent paper [2] where we established strong dynamical localization for a class of multi-particle Anderson models in a Euclidean space with an alloy-type random potential and a sub-exponentially decaying interaction of infinite range. We show that the localized eigenfunctions at low energies actually decay exponentially fast. This improves the results by Fauser and Warzel who established sub-exponential decay of eigenfunctions in presence of a sub-exponentially decaying interaction.
We establish strong dynamical localization for a class of multi-particle Anderson models in a Euclidean space with an alloy-type random potential and a sub-exponentially decaying interaction of infinite range. For the first time in the mathematical literature, the uniform decay bounds on the eigenfunction correlators at low energies are proved, in the multi-particle continuous configuration space, in the norm-distance and not in the Hausdorff pseudo-metric.
We assess the probability of resonances between sufficiently distant states in a combinatorial graph serving as the configuration space of an N-particle disordered quantum system. This includes the cases where the transition "shuffles" the particles in the configurations. In presence of a random external potential V, such pairs of configurations give rise to local (random) Hamiltonians which are strongly coupled, so that eigenvalue (or eigenfunction) correlator bounds are difficult to obtain. This difficulty, which occurs for three or more particles, results in eigenfunction decay bounds weaker than expected. We show that more efficient bounds, obtained so far only for 2-particle systems, extend to any number of particles.
We study multi-particle interactive quantum disordered systems on a polynomially-growing countable connected graph (Z,E). The novelty is to give localization bounds uniform in finite or infinite volumes (subgraphs) in Z^N as well as for the whole of Z^N. Such bounds are proved here by means of a comprehensive fixed-energy multi-particle multi-scale analysis. Another feature of the paper is that we consider -- for the first time in the literature -- an infinite-range (although fast-decaying) interaction between particles. For the models under consideration we establish (1) exponential spectral localization, and (2) strong dynamical localization with sub-exponential rate of decay of the eigenfunction correlators.
We present the first rigorous result on Anderson localization for interacting systems of quantum particles subject to a deterministic (e.g., almost periodic) disordered external potential. For a particular class of deterministic, fermionic, Anderson-type Hamiltonians on the lattice of an arbitrary dimension, and for a large class of underlying dynamical systems generating the external potential, we prove that the spectrum is pure point, all eigenstates are unimodal and feature a uniform exponential decay. In contrast to all prior mathematical works on multi-particle Anderson localization, we do not use the induction on the number of particles.
We study a particular class of families of multi-dimensional lattice Schrö\-dinger operators with deterministic (including quasi-periodic) potentials generated by the "hull" given by an orthogonal series over the Haar wavelet basis on the torus, of arbitrary dimension, with expansion coefficients considered as independent parameters. In the strong disorder regime, we prove Anderson localization for generic operator families, using a variant of the Multi-Scale Analysis, and show that all localized eigenfunctions are unimodal and feature uniform exponential decay away from their respective localization centers. Using the Klein--Molchanov argument and a variant of the Minami estimate for deterministic potentials, we prove the simplicity of the spectrum in our model. NOTE: This text completes our earlier manuscript (math-ph/0907.1494), originally uploaded in 2009 and revised in 2011, which is kept in \textbf{arXiv} in a reduced form, merely to avoid broken references in earlier works. Compared to [\texttt{math-ph/0907.1494}], we add the results on unimodality of the eigenstates, uniform dynamical localization, and simplicity of p.p. spectra. Compared to earlier versions of this preprint, the presentation has been adapted to the future extension of the main results (uniform localization, unimodality of the eigenfunctions) to the multi-particle Anderson Hamiltonians with a nontrivial interaction between the particles, which we plan to publish in a forthcoming paper.
We show that the hypothesis of regularity of the conditional distribution of the empiric average of a finite sample of IID random variables, given all the sample "fluctuations", which appeared in our earlier manuscript |1] in the context of the eigenvalue concentration analysis for multi-particle random operators, is satisfied for a class of probability distributions with piecewise-constant or sufficiently smooth probability density. It extends the well-known property of Gausssian IID samples.
We consider a particular class of lattice Schrödinger operators with deterministic potentials depending upon an infinite number of parameters in an auxiliary measurable space. We prove exponential dynamical localization for generic families in the strong disorder regime, using a variant of the Multi-Scale Analysis. In our model, the potential is generated by a function on a torus which is discontinuous (`haarsh') and constructed with the help of an expansion which reminds Haar's wavelet expansions.
This work is a continuation of \cite{C12b} where we described two elementary derivations of the variable-energy MSA bounds from their fixed-energy counterparts, in the framework of single-particle disordered quantum particle systems on graphs with polynomially bounded growth of balls. Here the approach of \cite{C12b} is extended to multi-particle Anderson Hamiltonians with interaction; it plays a role similar to that of the Simon--Wolf criterion for single-particle Hamiltonians. A simplified, fixed-energy multi-particle MSA scheme was developed in our earlier work \cite{C08a}, based on a multi-particle adaptation of techniques from Spencer's paper \cite{Sp88}. Combined with a simplified variant of the Germinet--Klein argument \cite{GK01} described in \cite{C12a}, the outcome of the fixed-energy analysis results in an elementary proof of multi-particle dynamical localization with the decay of eigenfunction correlators faster than any power-law.