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 consider the multi-particle Anderson model on the lattice with infinite range but sub-exponentially decaying interaction and show the Anderson localization consisting of the spectral exponential and the strong dynamical localization. In particular, the dynamical localization is proved int he Hilbert-Schmidt norm. The results concern very singular probability distributions such as the Bernoulli's measures.
We use a bootstrap argument to enhance the eigensystem multiscale analysis, introduced by Elgart and Klein for proving localization for the Anderson model at high disorder. The eigensystem multiscale analysis studies finite volume eigensystems, not finite volume Green's functions. It yields pure point spectrum with exponentially decaying eigenfunctions and dynamical localization. The starting hypothesis for the eigensystem bootstrap multiscale analysis only requires the verification of polynomial decay of the finite volume eigenfunctions, at some sufficiently large scale, with some minimal pro…
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.
In this paper, we study eigenvalue fluctuations of the finite volume Anderson model in the mesoscopic scale. We carry out this study in a regime of exponential localization and prove a central limit theorem for the eigenvalue counting function in a shrinking interval.
This work establishes the Anderson localization in both the spectral exponential and the strong dynamical localization for the multi-particle Anderson tight-binding model with correlated but strongly mixing random external potential. The results are obtained near the lower edge of the spectrum of the multi-particle Hamiltonian. In particular, the exponential decay of the eigenfunctions is proved in the max-norm and the dynamical localization in the Hilbert-Schmidt norm. The proofs need the conditional probability distribution function of the random external stochastic processes to obey the uni…
We study the solutions $u=u(x,t)$ to the Cauchy problem on $\mathbb Z^d\times(0,\infty)$ for the parabolic equation $\partial_t u=Δu+ξu$ with initial data $u(x,0)=1_{\{0\}}(x)$. Here $Δ$ is the discrete Laplacian on $\mathbb Z^d$ and $ξ=(ξ(z))_{z\in\mathbb Z^d}$ is an i.i.d.\ random field with doubly-exponential upper tails. We prove that, for large $t$ and with large probability, a majority of the total mass $U(t):=\sum_x u(x,t)$ of the solution resides in a bounded neighborhood of a site $Z_t$ that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamilton…
Under the weak interaction regime, we prove the one and the two volumes Wegner type bounds for one dimensional multi-particle models on the lattice and for very singular probability distribution functions such as the Bernoulli measures. The results imply the Anderson loclalization in both the spectral exponential and the strong dynamical localization for the one dimensional multi-particle Bernoulli-Anderson model with weak interaction.
In 1990, Klein, Lacroix, and Speis proved (spectral) Anderson localisation for the Anderson model on the strip of width $W \geqslant 1$, allowing for singular distribution of the potential. Their proof employs multi-scale analysis, in addition to arguments from the theory of random matrix products (the case of regular distributions was handled earlier in the works of Goldsheid and Lacroix by other means). We give a proof of their result avoiding multi-scale analysis, and also extend it to the general quasi-one-dimensional model, allowing, in particular, random hopping. Furthermore, we prove a …
We prove the occurrence of Anderson localisation for a system of infinitely many particles interacting with a short range potential, within the ground state Hartree-Fock approximation. We assume that the particles hop on a discrete lattice and that they are submitted to an external periodic potential which creates a gap in the non-interacting one particle Hamiltonian. We also assume that the interaction is weak enough to preserve a gap. We prove that the mean-field operator has exponentially localised eigenvectors, either on its whole spectrum or at the edges of its bands, depending on the str…
We consider the random Schrödinger operator on $\mathbb{R}$ obtained by perturbing the Laplacian with a white noise. We prove that Anderson localization holds for this operator: almost surely the spectral measure is pure point and the eigenfunctions are exponentially localized. We give two separate proofs of this result. We also present a detailed construction of the operator and relate it to the parabolic Anderson model. Finally, we discuss the case where the noise is smoothed out.
For the weakly interacting one-dimensional multi-particle Anderson model in the continuum space of configurations, we prove the spectral exponential and the strong dynamical localization. The results require the interaction amplitude to be sufficiently small. The general strategy uses the multi-scale analysis bounds. Actually, we show that the multi-scale analysis bounds of the single particle model remain stable when passing to multi-particle systems, provided that the inter-particle interaction is sufficiently small. The common probability distribution of the i.i.d. random external potential…
We establish the complete spectral exponential, and the strong Hilbert-Schmidt dynamical localization for the one-dimensional multi-particle Anderson tight-binding model and for weakly interacting particles system. In other words, we show stability of the one-dimensional localization from the single-particle to multi-particle systems with an arbitrary large but finite number of particles and for sufficient weakly interacting models. The proof uses the multi-scale analysis estimates for multi-particle systems. The common probability distribution function of the random external potential in the …
This paper concerns spectral properties of linear Schrödinger operators under oscillatory high-amplitude potentials on bounded domains. Depending on the degree of disorder, we prove the existence of spectral gaps amongst the lowermost eigenvalues and the emergence of exponentially localized states. We quantify the rate of decay in terms of geometric parameters that characterize the potential. The proofs are based on the convergence theory of iterative solvers for eigenvalue problems and their optimal local preconditioning by domain decomposition.
We study the multi-particle Anderson model in the continuum and show that under some mild assumptions on the random external potential and the inter-particle interaction, for any finite number of particles, the multi-particle lower edges of the spectrum are almost surely constant in absence of ergodicity. We stress that this result is not quite obvious and has to be handled carefully. In addition, we prove the spectral exponential and the strong dynamical localization of the continuous multi-particle Anderson model at low energy. The proof based on the multi-particle multi-scale analysis bound…
In this paper, we consider stochastic master equations describing the evolution of a multi-qubit system interacting with electromagnetic fields undergoing continuous-time measurements. By considering multiple z-type (Pauli z matrix on different qubits) and x-type (Pauli x matrix on all qubits) measurements and one control Hamiltonian, we provide general conditions on the feedback controller and the control Hamiltonian ensuring almost sure exponential convergence to a predetermined Greenberger-Horne-Zeilinger (GHZ) state, which is assumed to be a common eigenstate of the measurement operators. …
