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We consider classical acoustic waves in a medium described by a position dependent mass density %(x). We assume that %(x) is a random perturbation of a periodic function % 0 (x) and that the periodic acoustic operator A 0 = ?r 1 % 0 (x) r has a gap in the spectrum. We prove the existence of localized waves, i.e., nite energy solutions of the acoustic… (More)

We study acoustic and electromagnetic waves in a periodic medium (or any other background medium with a spectral gap) disturbed by a single defect, i.e., a local disturbance analogous to a well potential in solid state physics. We show that defects do not change the essential spectrum of the associated nonnegative operators and can only create isolated… (More)

We give a simple, transparent, and intuitive proof that all eigenvalues of the Anderson model in the region of localization are simple. The Anderson tight binding model is given by the random Hamiltonian H ω = −∆ + V ω on 2 (Z d), where ∆(x, y) = 1 if |x − y| = 1 and zero otherwise, and the random potential V ω = {V ω (x), x ∈ Z d } consists of independent… (More)

- Jean-Michel Combes, Franç Ois Germinet, Abel Klein
- 2008

We generalize Minami's estimate for the Anderson model and its extensions to n eigenvalues, allowing for n arbitrary intervals and arbitrary single-site probability measures with no atoms. As an application, we derive new results about the multiplicity of eigenvalues and Mott's formula for the ac-conductivity when the single site probability distribution is… (More)

- François Germinet, Abel Klein
- 2001

We investigate the Anderson metal-insulator transition for random Schrödinger operators. We define the strong insulator region to be the part of the spectrum where the random operator exhibits strong dynamical localization in the Hilbert-Schmidt norm. We introduce a local transport exponent β(E), and set the metallic transport region to be the part of the… (More)

- Henrique Von Dreifus, Abel Klein, J Fernando Perez
- 1994

We prove innnite diierentiability of the magnetization and of all quenched correlation functions for disordered spin systems at high temperature or strong magnetic eld in the presence of Griiths' singularities. We also show uniqueness of the Gibbs state and exponential decay of truncated correlation functions with probability one. Our results are obtained… (More)

We consider lattice versions of Maxwell's equations and of the equation that governs the propagation of acoustic waves in a random medium. The vector nature of electromagnetic waves is fully taken into account. The medium is assumed to be a small perturbation of a periodic one. We prove rigorously that localized eigenstates arise in a vicinity of the edges… (More)

We consider electromagnetic waves in a medium described by a position dependent dielectric constant "(x). We assume that "(x) is a random perturbation of a periodic function " 0 (x) and that the periodic Maxwell operator M 0 = r 1 " 0 (x) r has a gap in the spectrum, were r = rr. We prove the existence of localized waves, i.e., nite energy solutions of… (More)

- Abel Klein, Alexander Figotin
- SIAM Journal of Applied Mathematics
- 1998

- Abel Klein
- 1994

We prove that the Anderson Hamiltonian H = ? + V on the Bethe Lattice has \extended states" for small disorder. More precisely, given any closed interval I contained in the interior of the spectrum of the Laplacian on the Bethe lattice, we prove that for small disorder H has purely absolutely continuous spectrum in I with probability one (i.e., ac (H)\I = I… (More)