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We show almost sure simplicity of eigenvalues for several models of Anderson-type random Schr\"odinger operators, extending methods introduced by Simon for the discrete Anderson model. These methods work throughout the spectrum and are not restricted to the localization regime. We establish general criteria for the simplicity of eigenvalues which can be interpreted as separately excluding the absence of local and global symmetries, respectively. The criteria are applied to Anderson models with matrix-valued potential as well as with single-site potentials supported on a finite box., Comment: 20 pages

This work is a contribution to the theory of the absolutely continuous spectrum of one-dimensional Schriidinger operators, i.e., Sturm-Liouville operators generated by differential expressions of the form ru = --u” + qu. In proofs of absolute continuity for such operators and also for the closely related Dirac systems always asymptotic properties of solutions of zu = lu for real or complex values of 1 have played an important role, see, for example, [18, 19, 8, 4, 9, lo]. In these works and a number of others particular classes of potentials q were treated using properties of solutions in the course of the proof of absolute continuity. Somewhat more recently the connection between asymptotic properties of solutions and absolute continuity was given a theory, i.e., results of the following form were proved: Let the solutions of zu = lu satisfy some specific asymptotic property for A varying in some subset of R (no or almost no a priori assumptions on q), then operators generated by z are absolutely continuous in this set. A first result of this type was given by Carmona [3], who proved that local uniform boundedness of solutions together with their derivatives in some open A-set implies pure absolute continuity in this set. Here we will mostly rely on the concept of (non-).&or&racy of solutions, which was shown by Gilbert and Pearson [7, 12, 61 to be a powerful tool in the investigation of the continuous spectrum of Sturm-Liouville operators. In Section 2 we give some consequences of their results, whi_ch are useful in proofs of absolute continuity. These consequences range from non-existence of imbedded eigenvalues and the existence of absolutely continuous parts to pure absolute continuity and the existence of transient 210 0022-247x/92 $5.00

This paper establishes dynamical localization properties of certain families of unitary random operators on the d-dimensional lattice in various regimes. These operators are generalizations of one-dimensional physical models of quantum transport and draw their name from the analogy with the discrete Anderson model of solid state physics. They consist in a product of a deterministic unitary operator and a random unitary operator. The deterministic operator has a band structure, is absolutely continuous and plays the role of the discrete Laplacian. The random operator is diagonal with elements given by i.i.d. random phases distributed according to some absolutely continuous measure and plays the role of the random potential. In dimension one, these operators belong to the family of CMV-matrices in the theory of orthogonal polynomials on the unit circle. We implement the method of Aizenman-Molchanov to prove exponential decay of the fractional moments of the Green function for the unitary Anderson model in the following three regimes: In any dimension, throughout the spectrum at large disorder and near the band edges at arbitrary disorder and, in dimension one, throughout the spectrum at arbitrary disorder. We also prove that exponential decay of fractional moments of the Green function implies dynamical localization, which in turn implies spectral localization. These results complete the analogy with the self-adjoint case where dynamical localization is known to be true in the same three regimes., 62 pages

Bounds on the exponential decay of generalized eigenfunctions of bounded and unbounded selfadjoint Jacobi matrices are established. Two cases are considered separately: (i) the case in which the spectral parameter lies in a general gap of the spectrum of the Jacobi matrix and (ii) the case of a lower semi-bounded Jacobi matrix with values of the spectral parameter below the spectrum. It is demonstrated by examples that both results are sharp. We apply these results to obtain a "many barriers-type" criterion for the existence of square-summable generalized eigenfunctions of an unbounded Jacobi matrix at almost every value of the spectral parameter in suitable open sets. As an application, we provide examples of unbounded Jacobi matrices with a spectral mobility edge., This is a substantially revised and expanded version of 0711.4035v1

Methods of Spectral Analysis in Mathematical Physics : Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2006, Lund, Sweden

We give a proof of dynamical localization in the form of exponential decay of spatial correlations in the time evolution for the one-dimensional continuum Anderson model via the fractional moments method. This follows via exponential decay of fractional moments of the Green function, which is shown to hold at arbitrary energy and for any single-site distribution with bounded, compactly supported density., Comment: 13 pages

We consider the isotropic or anisotropic XY spin chain in the presence of a transversal random magnetic field, with parameters given by random variables. It is shown that eigenfunction correlator localization of the corresponding effective one-particle Hamiltonian implies a uniform area law bound in expectation for the bipartite entanglement entropy of all eigenstates of the XY chain, i.e. a form of many-body localization at all energies. Here entanglement with respect to arbitrary connected subchains of the chain can be considered. Applications where the required eigenfunction correlator bounds are known include the isotropic XY chain in random field as well as the anisotropic chain in strong or strongly disordered random field., 28 pages, v2: changes in introduction after feedback received, some references added, v3: some improvements and updates, main results unchanged

BACKGROUND: Bone marrow edema (BME) is a common cause of pain of the musculoskeletal system. The aim of the study was to assess the efficacy of iloprost in the treatment of BME of different localizations and etiologies. PATIENTS AND METHODS: We reviewed 104 patients (54 male, 50 female) with BME. Their mean age was 52.8 ± 14.7 years. BME was located 50 times in the knee, 19 times in the talus, 18 times in the femoral head and 17 times in other bones. Patients were allocated to three distinct etiological groups: 27 cases were estimated to have idiopathic BME, 16 post-traumatic BME and the other 61 BME secondary to activated osteoarthritis or mechanical stress. Therapy consisted of a series of five iloprost infusions with either 20, 25 or 50 μg of iloprost given over 6 hours on 5 consecutive days each. RESULTS: At the clinical follow-up four months after therapy, the pain level of the 104 patients at rest had diminished by a mean of 73% (p < 0.0001): 64% of patients reported a reduction, 34% no change and 2% an increase in pain at rest. Pain under stress decreased by a mean of 59%, (p < 0.0001): 76% of patients had less pain during activity, 22% no change from baseline and 2% an increased pain level. On MRI, 65% had significant reduction of BME size or complete normalization and 20% showed no change. Worsening of the MRI pattern was found in 2%. 13% were lost to MRI follow-up. Side effects were significantly reduced by lowering the daily dose from 50 to 20 μg, without impairment of therapeutic effect. CONCLUSION: The authors conclude that the use of parenteral iloprost might be a viable method in the treatment of BME of different etiologies.

We consider unitary analogs of $1-$dimensional Anderson models on $l^2(\Z)$ defined by the product $U_\omega=D_\omega S$ where $S$ is a deterministic unitary and $D_\omega$ is a diagonal matrix of i.i.d. random phases. The operator $S$ is an absolutely continuous band matrix which depends on a parameter controlling the size of its off-diagonal elements. We prove that the spectrum of $U_\omega$ is pure point almost surely for all values of the parameter of $S$. We provide similar results for unitary operators defined on $l^2(\N)$ together with an application to orthogonal polynomials on the unit circle. We get almost sure localization for polynomials characterized by Verblunski coefficients of constant modulus and correlated random phases.

Inventiones mathematicae, 163 (2), ISSN:0020-9910, ISSN:1432-1297

The fractional moment method, which was initially developed in the discrete context for the analysis of the localization properties of lattice random operators, is extended to apply to random Schr\"odinger operators in the continuum. One of the new results for continuum operators are exponentially decaying bounds for the mean value of transition amplitudes, for energies throughout the localization regime. An obstacle which up to now prevented an extension of this method to the continuum is the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. This difficulty is resolved through an analysis of the resonance-diffusing effects of the disorder., Comment: This is a brief announcement of math-ph/0308023, written for the proceedings of ICMP 2003

We use scattering theoretic methods to prove strong dynamical and exponential localization for one dimensional, continuum, Anderson-type models with singular distributions; in particular the case of a Bernoulli distribution is covered. The operators we consider model alloys composed of at least two distinct types of randomly dispersed atoms. Our main tools are the reflection and transmission coefficients for compactly supported single site perturbations of a periodic background which we use to verify the necessary hypotheses of multi-scale analysis. We show that non-reflectionless single sites lead to a discrete set of exceptional energies away from which localization occurs., Comment: 32 pages

We study one-dimensional, continuum Bernoulli-Anderson models with general single-site potentials and prove positivity of the Lyapunov exponent away from a discrete set of critical energies. The proof is based on Furstenberg’s Theorem. The set of critical energies is described explicitly in terms of the transmission and reflection coefficients for scattering at the single-site potential. In examples we discuss the asymptotic behavior of generalized eigenfunctions at critical energies.

The past two decades have brought widespread interest and a large number of publications in the mathematical theory of random operators. While this has lead to many deep results and considerable mathematical insights, there are still important open problems which are far from being solved. This becomes particularly clear when looking at what is known for models other than the relatively well understood Anderson model and its generalizations. Of course, there are big challenges left in the Anderson model as well (delocalization!), but for other physically significant models (e.g. Poisson model, random displacement model) even the seemingly better understood phenomenon of localization is mathematically far from being settled.

We prove an upper bound proportional to the surface area for the bipartite entanglement of the ground state and thermal states of harmonic oscillator systems with disorder, as measured by the logarithmic negativity. Our assumptions are satisfied for some standard models that are almost surely gapless in the thermodynamic limit.

In the quantum mechanical one-electron approximation a solid consisting of nuclei producing single site potentials f i at points x i ∈ ℝ d is described by the Schrodinger operator H = -Δ + Σ i f i (x - x i ). Random media are modeled by introducing suitable random parameters into H.

We give trace norm estimates for products of integral operators and for diffusion semigroups. These are applied to differences of heat semigroups. A natural example of an integral operator with finite trace which is not trace class is given.

We study a unitary version of the one-dimensional Anderson model, given by a five diagonal deterministic unitary operator multiplicatively perturbed by a random phase matrix. We fully characterize positivity and vanishing of the Lyapunov exponent for this model throughout the spectrum and for arbitrary distributions of the random phases. This includes Bernoulli distributions, where in certain cases a finite number of critical spectral values, with vanishing Lyapunov exponent, exists. We establish similar results for a unitary version of the random dimer model., Comment: 18 pages

Let a quantum mechanical system be in the normalized state f; measuring the observable quantity which is represented by H, we have the probability I~.l 2 to find the value a(n) and the probability density to find a(2) in the continuous spectrum is Ic~(2)l 2. b) Let for example H = A + V be a one-body Hamiltonian (i.e. V(x) ~ 0 for [xJ--.oo). We expect the T~ to be bounded or at most slowly increasing (plane waves for V=0). On the other hand, for values E$a(H) the solutions T of (--A + V) T = E 7 ~ should grow fast (exponentially) at infinity. These conjectures can be summarized to

We prove exponential localization at all energies for one-dimensional continuous Anderson-type models with single site potentials of changing sign. A periodic background potential is allowed. The main problem arises from non-monotonicity; i.e., the operator does not depend monotonically in the form sense on the random parameters. We show that the method of “two-parameter spectral averaging,” recently devised by Buschmann and Stolz to prove localization for Poisson and random displacement models, can be modified to work for the type of Anderson model considered here.

Our goal is to show that large classes of Schrodinger operators H =− Δ + V in L 2 ( R d ) exhibit intervals of dense pure point spectrum, in any dimension d . We approach this by assuming that the potential V ( x ) coincides with a potential V 0 ( x ) of a “comparison operator” H 0 =− Δ + V 0 on a sequence of ring shaped (but nog necessarily spherical) regions U n , n =1, 2, … . For energies in the resolvent set ρ ( H 0 ) of H 0 the regions U n act as “effective barriers” in the sense of quantum mechanical scattering under the potential V . Under certain assumptions on the geometry of the U n and their complements we show that (i) σ ac ( H ( λ ))∩ ρ ( H 0 )=∅ for every λ ∈ R , and (ii) σ c ( H ( λ ))∩ ρ ( H 0 )=∅ for almost every λ ∈ R with respect to Lebesgue measure. Here σ ac and σ c denote the absolutely continuous and continuous spectrum, respectively, and H ( λ ) is a “local randomization” of H , i.e., H ( λ )= H + λW , where W is any continuous and compactly supported perturbation of fixed sign. Our assumptions leave plenty of room for examples where the spectrum of H fills entire spectral gaps of H 0 . This leads to intervals of dense pure point spectrum for H ( λ ). We also give an explicit decay estimate for eigenfunctions, thus establishing localization for H ( λ ) in arbitrary spectral gaps of H 0 .

We use elementary methods to give a full characterization of the spectral properties of unbounded Jacobi matrices with zero diagonal and off-diagonal entries of the type λn=nα+cn, where 12