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We consider a random Schr\"odinger operator on the binary tree with a random potential which is the sum of a random radially symmetric potential, $Q_r$, and a random transversally periodic potential, $\kappa Q_t$, with coupling constant $\kappa$. Using a new one-dimensional dynamical systems approach combined with Jensen's inequality in hyperbolic space (our key estimate) we obtain a fractional moment estimate proving localization for small and large $\kappa$. Together with a previous result we therefore obtain a model with two Anderson transitions, from localization to delocalization and back to localization, when increasing $\kappa$. As a by-product we also have a partially new proof of one-dimensional Anderson localization at any disorder., Comment: 25 pages, 1 figure

We study low-energy properties of the random displacement model, a random Schr\"odinger operator describing an electron in a randomly deformed lattice. All periodic displacement configurations which minimize the bottom of the spectrum are characterized. While this configuration is essentially unique for dimension greater than one, there are infinitely many different minimizing configurations in the one-dimensional case. The latter leads to unusual low energy asymptotics for the integrated density of states of the one-dimensional random displacement model. For symmetric Bernoulli-distributed displacements it has a $1/\log^2$-singularity at the bottom of the spectrum. In particular, it is not H\"older-continuous., Comment: 14 pages, 1 figure

We provide a characterization of the spectral minimum for a random Schr\"odinger operator of the form $H=-\Delta + \sum_{i \in \Z^d}q(x-i-\omega_i)$ in $L^2(\R^d)$, where the single site potential $q$ is reflection symmetric, compactly supported in the unit cube centered at 0, and the displacement parameters $\omega_i$ are restricted so that adjacent single site potentials do not overlap. In particular, we show that a minimizing configuration of the displacements is given by a periodic pattern of densest possible $2^d$-clusters of single site potentials. The main tool to prove this is a quite general phenomenon in the spectral theory of Neumann problems, which we dub ``bubbles tend to the boundary.'' How should a given compactly supported potential be placed into a bounded domain so as to minimize or maximize the first Neumann eigenvalue of the Schr\"odinger operator on this domain? For square or rectangular domains and reflection symmetric potentials, we show that the first Neumann eigenvalue is minimized when the potential sits in one of the corners of the domain and is maximized when it sits in the center of the domain. With different methods we also show a corresponding result for smooth strictly convex domains., Comment: 18 pages, 4 figures

For a large class of multi-dimensional Schrodinger operators it is shown that the absolutely continuous spectrum is essentially supported by [ 0 , ∞ ) . We require slow decay and mildly oscillatory behavior of the potential in a cone and can allow for arbitrary non-negative bounded potential outside the cone. In particular, we do not require the existence of wave operators. The result and method of proof extends previous work by Laptev, Naboko and Safronov.

We prove the existence of ballistic transport for the Schr\"odinger operator with limit-periodic or quasi-periodic potential in dimension two. This is done under certain regularity assumptions on the potential which have been used in prior work to establish the existence of an absolutely continuous component and other spectral properties. The latter include detailed information on the structure of generalized eigenvalues and eigenfunctions. These allow to establish the crucial ballistic lower bound through integration by parts on an appropriate extension of a Cantor set in momentum space, as well as through stationary phase arguments., Comment: 30 pages, more discussion and motivation of main result and assumptions added, final version, to appear in Comm.Math.Phys

We present a short, new, self-contained proof of localization properties of multi-dimensional continuum random Schodinger operators in the fluctuation boundary regime. Our method is based on the recent extension of the fractional moment method to continuum models in [2] but does not require the random potential to satisfy a covering condition. Applications to random surface potentials and potentials with random displacements are included.

We prove quantum dynamical lower bounds for one-dimensional continuum Schrodinger operators that possess critical energies for which there is slow growth of transfer matrix norms and a large class of compactly supported initial states. This general result is applied to a number of models, including the Bernoulli–Anderson model with a constant single-site potential.

We consider Schr\"odinger operators in $\ell^2(\Z)$ whose potentials are obtained by randomly concatenating words from an underlying set $\mathcal{W}$ according to some probability measure $\nu$ on $\mathcal{W}$. Our assumptions allow us to consider models with local correlations, such as the random dimer model or, more generally, random polymer models. We prove spectral localization and, away from a finite set of exceptional energies, dynamical localization for such models. These results are obtained by employing scattering theoretic methods together with Furstenberg's theorem to verify the necessary input to perform a multiscale analysis., Comment: 19 pages

Recent results on localization, both exponential and dynamical, for various models of one-dimensional, continuum, random Schrodinger operators are reviewed. This includes Anderson models with indefinite single site potentials, the BernoulliAnderson model, the Poisson model, and the random displacement model. Among the tools which are used to analyse these models are generalized spectral averaging techniques and results from inverse spectral and scattering theory. A discussion of open problems is included.

We prove exponential localization at all energies for two types of one-dimensional random Schrödinger operators: the Poisson model and the random displacement model. As opposed to Anderson-type models, these operators are not monotonic in the random parameters. Therefore the classical one-parameter version of spectral averaging, as used in localization proofs for Anderson models, breaks down. We use the new method of two-parameter spectral averaging and apply it to the Poisson as well as the displacement case. In addition, we apply results from inverse spectral theory, which show that two-parameter spectral averaging works for sufficiently many energies (all but a discrete set) to conclude localization at all energies.

We use scattering theoretic methods to prove exponential localization for random displacement models in one dimension. The operators we consider model both quantum and classical wave propagation. Our main tools are the reflection and transmission coefficients for compactly supported single site perturbations. We show that randomly displaced, non-reflectionless single sites lead to localization.

We prove pure point spectrum with exponentially decaying eigenfunctions at all band edges for Schrodinger Operators with a periodic potential plus a random potential of the form V w (x) = Σq i (w)f(x - i), where $f$ decays at infinity like |x|−m for m>4d resp. $m>3d depending on the regularity of f. The random variables q i are supposed to be independent and identically distributed. We assume that their distribution has a bounded density of compact support.

The absolutely continuous and singular spectrum of one-dimensional Schrodinger operators with slowly oscillating potentials and perturbed periodic potentials is studied, continuing similar investigations for Jacobi matrices from [14]. Trace class methods are used to locate the singular spectrum. The absolutely continuous spectrum is determined by iterated diagonalization of transfer matrices to prove boundedness of eigensolutions.

Let A be a self-adjoint operator defined by a general singular ordinary differential expression τ on an interval (a, b), − ∞ ≤ a < b ≤ ∞. We show that isolated eigenvalues in any gap of the essential spectrum of A are exactly the limits of eigenvalues of suitably chosen self-adjoint realizations An of τ on subintervals (an, bn) of (a, b) with an → a, bn → b. This means that eigenvalues of singular ordinary differential operators can be approximated by eigenvalues of regular operators.

For Jacobi matrices on the discrete half line with slowly oscillating potentials the absolutely continuous and singular spectrum is located. The results, which can be extended to perturbed periodic potentials, show that separated regions of purely absolutely continuous resp. purely singular spectrum appear. The main tools in the proof of absolute continuity are the method of subordinacy and an abstract result on the iterated diagonalization of products of 2×2-matrices, which is applied to transfer matrices. The singular spectrum is found by using a result of Simon and Spancer.

For Sturm-Liouville operators on the half line, we show that the property of having singular, singular continuous, or pure point spectrum for a set of boundary conditions of positive measure depends only on the behavior of the potential at infinity. We also prove that existence of recurrent spectrum implies that of singular spectrum and that “almost sure” existence of L_2-solutions implies pure point spectrum for almost every boundary condition. The same results hold for Jacobi matrices on the discrete half line.

We investigate spectral properties of a discrete random displacement model, a Schr\"odinger operator on $\ell^2(\Z^d)$ with potential generated by randomly displacing finitely supported single-site terms from the points of a sublattice of $\Z^d$. In particular, we characterize the upper and lower edges of the almost sure spectrum. For a one-dimensional model with Bernoulli distributed displacements, we can show that the integrated density of states has a $1/\log^2$-singularity at external as well as internal band edges., Comment: 26 pages, 2 figures

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

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

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.

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 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.

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