A metal-insulator transition for the almost Mathieu model

Journal of Functional Analysis, 1982

For a dense G, of pairs (,I, a) in R', we prove that the operator (Hu)(n) = u(n + 1) + u(n -1) + I cos(2nan + 0) u(n) has a nowhere dense spectrum. Along the way we prove several interesting results about the case a =p/q of which we mention: (a) If qB is not an integral multiple of A, then all gaps are open, and (b) If q is even and 0 is chosen suitably, then the middle gap is closed for all I..

arXiv (Cornell University), 2015

We generalize the approach to localization in one dimension introduced by Kunz-Souillard, and refined by Delyon-Kunz-Souillard and Simon, in the early 1980's in such a way that certain correlations are allowed. Several applications of this generalized Kunz-Souillard method to almost periodic Schrödinger operators are presented. On the one hand, we show that the Schrödinger operators on ℓ 2 (Z) with limit-periodic potential that have pure point spectrum form a dense subset in the space of all limit-periodic Schrödinger operators on ℓ 2 (Z). More generally, for any bounded potential, one can find an arbitrarily small limit-periodic perturbation so that the resulting operator has pure point spectrum. Our result complements the known denseness of absolutely continuous spectrum and the known genericity of singular continuous spectrum in the space of all limit-periodic Schrödinger operators on ℓ 2 (Z). On the other hand, we show that Schrödinger operators on ℓ 2 (Z) with arbitrarily small one-frequency quasi-periodic potential may have pure point spectrum for some phases. This was previously known only for one-frequency quasiperiodic potentials with • ∞ norm exceeding 2, namely the super-critical almost Mathieu operator with a typical frequency and phase. Moreover, this phenomenon can occur for any frequency, whereas no previous quasi-periodic potential with Liouville frequency was known that may admit eigenvalues for any phase.

We discuss criteria for a self-adjoint operator on L 2 (X) to have empty essential spectrum. We state a general result for the case of a locally compact abelian group X and give examples for X = R n. 1. Let ∆ be the positive Laplacian on R n. We set Ba(r) = {x ∈ R n | |x − a | ≤ r} and Ba = Ba(1). Proposition 1. Let V be a real locally integrable function on R n such that: (i) if λ> 0 then the measure ωλ of the set {x ∈ Ba | V (x) < λ} satisfies lima→ ∞ ωλ(a) = 0, (ii) the negative part of V satisfies V − ≤ µ ∆ + ν for some positive real numbers µ, ν with 0 < µ < 1. Then the spectrum of the self-adjoint operator H associated to the form sum ∆ + V is purely discrete. Remark 2. Let V ± = max{±V, 0} and for each λ> 0 let Ωλ = {x | V+(x) < λ}. Then ωλ(a) is the measure of the set Ba ∩ Ωλ. From Lemma 5 it follows that the condition (i) is equivalent to dx lim

Duke Mathematical Journal, 1983

1. Introduction. In this paper, we will study Schr6dingeroperators, -A + V, on L2(R) where V is an almost periodic function on R". We will be especially interested in the case t, where we will also consider the finite difference analog on/2(Z) ... (MU)n= u,,+, + u,,_ + V(n)u

Asymptotic Analysis

We consider a 2D Schroedinger operator H0 with constant magnetic field, on a strip of finite width. The spectrum of H0 is absolutely continuous, and contains a discrete set of thresholds. We perturb H0 by an electric potential V which decays in a suitable sense at infinity, and study the spectral properties of the perturbed operator H = H0 + V . First, we establish a Mourre estimate, and as a corollary prove that the singular continuous spectrum of H is empty, and any compact subset of the complement of the threshold set may contain at most a finite set of eigenvalues of H, each of them having a finite multiplicity. Next, we introduce the Krein spectral shift function (SSF) for the operator pair (H,H0). We show that this SSF is bounded on any compact subset of the complement of the threshold set, and is continuous away from the threshold set and the eigenvalues of H. The main results of the article concern the asymptotic behaviour of the SSF at the thresholds, which is described in ...

XVIth International Congress on Mathematical Physics, 2010

We study a C r nearly integrable Hamiltonian system Hε(q, p) = 1 2 p, p + H 1 (q, p) defined on T 3 × R 3. Let Σ = {(q, p) : Hε(q, p) = 1 2 } and µ Σ 1 be the restriction of Lebesgue measure on T 3 × R 3 to Σ. We prove there is a perturbation H 1 (q, p) ∈ C r , H 1 C r ≤ 1 and an orbit (q(t), p(t)) : R → T 3 × R 3 of the Hamiltonian equation {q = ∂pHε,ṗ = −∂qHε} such that µ Σ (S t∈R (q(t), p(t))) ≥ 1 2 .

Journal of Mathematical Analysis and Applications, 1990

Integral Equations and Operator Theory, 1997

An explicit derivation of a tridiagonal matrix form for the almost Mathieu operator (Harper's equation) is obtained via conjugation with a reflection operator, valid for all rational values of the rotation parameter. The difference between even and odd values of the denominator is highlighted. This tridiagonal form is useful for numerical eigenvalue computations; some Matlab code is included. Lamoureux o.o

Reviews in Mathematical Physics, 2014

Let H 0,D (resp., H 0,N ) be the Schrödinger operator in constant magnetic field on the half-plane with Dirichlet (resp., Neumann) boundary conditions, and let H ℓ := H 0,ℓ − V , ℓ = D, N , where the scalar potential V is non negative, bounded, does not vanish identically, and decays at infinity. We compare the distribution of the eigenvalues of H D and H N below the respective infima of the essential spectra. To this end, we construct effective Hamiltonians which govern the asymptotic behaviour of the discrete spectrum of H ℓ near inf σ ess (H ℓ ) = inf σ(H 0,ℓ ), ℓ = D, N . Applying these Hamiltonians, we show that σ disc (H D ) is infinite even if V has a compact support, while σ disc (H N ) could be finite or infinite depending on the decay rate of V .

2020

We study sufficient conditions for the absence of positive eigenvalues of magnetic Schrodinger operators in $\mathbb{R}^d,\, d\geq 2$. In our main result we prove the absence of eigenvalues above certain threshold energy which depends explicitly on the magnetic and electric field. A comparison with the examples of Miller--Simon shows that our result is sharp as far as the decay of the magnetic field is concerned. As applications, we describe several consequences of the main result for two-dimensional Pauli and Dirac operators, and two and three dimensional Aharonov--Bohm operators.