What is Localization?

Abstract

We examine various issues relevant to localization in the Anderson model. We show there is more to localization than exponentially localized states by presenting an example with such states but where (x(t) )/t is unbounded for any 6 ) 0. We show that the recently discovered instability of localization under rank one perturbations is only a weak instability.(H u) (n) = g u(n + j ) + V~(n)u(n), I jl=& where the potentials V are identically distributed independent random variables with distribution [2] 1 2n g( ""](x) dx, with g~""t he characteristic function of the interval Many of the claimed proofs of localization show that, for almost all cu, an Anderson model Hamiltonian H has a complete set of normalized eigenvectors [3] (p" obeying (n) (where A is fixed, the n 's are some centers of localization, and the C 's are constants depending on~and m.Our first result is an example that shows that mere "exponential localization' of eigenfunctions in the form of Eq. (2) need not have very strong consequences for the dynamics. We can construct a nonrandom potential V in one dimension with the following: (i) H has a complete set of normalized eigenvectors obeying Eq. (2). (ii) Let (x ) (t) denote (e " Bo, x e " Bo); then for any 6 ) 0,The potential U for this example is PACS numbers: 72. 15.Rn Localization in random media is basic to a variety of physical situations. We wish to report here on a number of rigorous mathematical results that shed light on the phenomenon of localization in the Anderson model. Mathematically complete proofs of our results will appear elsewhere [1]. Our goal here is to describe the ideas behind the results. Throughout, we will consider the Anderson model, that is, the Hamiltonian H on 4 (Z") (namely, on the ddimensional cubic lattice) V(n) = 3 cos(27rnn + 0) + AB"o, ]p (n)[~C e'~"-"' eCondition (3) says that the constants C"of(2) are allowed to grow at a rate which is less than exponential in the distance of the n"'sfrom the origin. SULE is closely related to a dynamical condition, which we call semiuniform dynamical localization (SUDL) (n g) ( We have proven that (3) implies (4) with A arbitrarily close to A, and that if H has simple eigenvalues [6], which we consider on the positive half of the lattice (n0 ), with a Dirichlet (or any other) boundary condition at the origin. The 3 in front of the cosine can be replaced by any number larger than 2, and is chosen so that when A = 0 the problem has a positive Lyapunov exponent [5].The n is an irrational, which is specially chosen so that for suitable time scales T"~~, U is so close to periodic that we can show (x (T")) is large compared to T2The local perturbation A6 o pushes the spectrum to be pure point and forces Eq. (2) to hold. While V is very far from random, it illustrates that Eq. (2) is not enough to restrict dynamics. The main failing in (2) is the total freedom given to the constants C . Indeed, when one thinks of "localization, " one usually thinks of the eigenvectors as being confined, at least roughly, within some typical length s...

On the Problem of Dynamical Localization in the Nonlinear Schrödinger Equation with a Random Potential

“…where ε > 0, x m is the localization center defined by the point where u m (x) has its maximum, γ is the inverse of the localization length, ξ = γ −1 , and D ω,ε is a constant dependent on ε and the realization of the disordered potential [50,51] (better estimates were proven recently in [52,53]). It is of importance that D ω,ε does not depend on the energy of the state.…”

“…ε is an average which does not include the bad realizations of the potentials. Because of exponential localization [48][49][50][51] 25) with 0 < c < s · min E γ (E) . The parameter q can be made arbitrarily close to 1, therefore in our case (3.22) holds for 0 < s < 1 − η with η arbitrarily small.…”

A Short Introduction to Anderson Localization

“…If the C ω,n are allowed to arbitrarily grow, in n, then, in fact, the eigenvectors can be extended over arbitrarily large length scales, possibly leading to transport arbitrarily close to the ballistic motion, even though one has only pure point spectrum. This is nicely discussed in del Rio et al (1995) with the proofs given in del Rio et al (1996).…”

The stochastic Mathieu's equation

“…Deych et al (1998) solved this problem numerically using Monte Carlo simulations and computed the Lyapunov exponents to determine the stability of waves moving through a random medium. The effect of the randomness can diffuse the incoming wave, as we might expect, however, it can also be strengthened, a process known as Anderson localization (del Rio et al 1995). This process was first discovered in quantum mechanical systems but it has since been determined that it is relevant for many other types of waves.…”