Elisabetta Scoppola

Constructive proof of localization in the Anderson tight binding model

We prove that, for large disorder or near the band tails, the spectrum of the Anderson tight binding Hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues. The corresponding eigenfunctions are exponentially well localized. These results hold in arbitrary dimension and with probability one. In one dimension, we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder. Our techniques extend to other physical systems which exhibit localization phenomena, such as infinite systems of coupled harmonic oscillators, or random Schrödinger operators in the continuum.

Spectral properties of one dimensional quasi-crystals

AbstractIn this paper we prove that the one dimensional Schrödinger operator onl2(ℤ) with potential given by:

Markov Chains with Exponentially Small Transition Probabilities: First Exit Problem from a General Domain. I. The Reversible Case

\upsilon (n) = \lambda \chi _{[1 - \alpha , 1[} (x + n\alpha )\alpha \notin \mathbb{Q}

Renormalization Group for Markov Chains and Application to Metastability

has a Cantor spectrum of zero Lebesgue measure for any irrationalα and any λ>0. We can thus extend the Kotani result on the absence of absolutely continuous spectrum for this model, to all % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepGe9fr-xfr-x% frpeWZqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhaiiaacq% WFiiIZtuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbbiab% +nj8ubaa!4628!\[x \in \mathbb{T}

On the Swendsen-Wang dynamics. I. Exponential convergence to equilibrium

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Small random perturbations of dynamical systems: Exponential loss of memory of the initial condition

We propose a new approach for the estimate of the rate of degeneracy of the lowest eigenvalues of the Schrödinger operator in the presence of tunneling based on the theory of diffusion processes. Our method provides lower and upper bounds for the energy splittings and the rates of localization of the wave functions and enables us to discuss cases which, as far as we know, have never been treated rigorously in the literature. In particular we give an analysis of the effect on eigenvalues and eigenfunctions of localized deformations of 1) symmetric double well potentials 2) potentials periodic and symmetric over a finite interval. Theses situations are characterized by a remarkable dependence on such deformations. Our probabilistic techniques are inspired by the theory of small random perturbations of dynamical systems.

Metastability for Markov Chains: A General Procedure Based on Renormalization Group Ideas

In this paper we consider aperiodic ergodic Markov chains with transition probabilities exponentially small in a large parameter β. We extend to the general, not necessarily reversible case the analysis, started in part I of this work, of the first exit problem from a general domainQ containing many stable equilibria (attracting equilibrium points for the β=∞ dynamics). In particular we describe the tube of typical trajectories during the first excursion outsideQ.