Günter Stolz

Moment analysis for localization in random Schrödinger operators

We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonance-diffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weak-L1 estimate concerning the boundary-value distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.

Localization for random perturbations of periodic Schrödinger operators

We prove localization for Anderson-type random perturbations of periodic Schr dinger operators on R near the band edges. General, possibly unbounded, single site potentials of fixed sign and compact support are allowed in the random perturbation. The proof is based on the following methods: (i) A study of the band shift of periodic Schr dinger operators under linearly coupled periodic perturbations. (ii) A proof of the Wegner estimate using properties of the spatial distribution of eigenfunctions of finite box hamiltonians. (iii) An improved multiscale method together with a result of de Branges on the existence of limiting values for resolvents in the upper half plane, allowing for rather weak disorder assumptions on the random potential. (iv) Results from the theory of generalized eigenfunctions and spectral averaging. The paper aims at high accessibility in providing details for all the main steps in the proof.

Bounded solutions and absolute continuity of Sturm-Liouville operators

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

Delocalization in random polymer models

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Localization for one-dimensional, continuum, Bernoulli-Anderson models

Abstract: A random polymer model is a one-dimensional Jacobi matrix randomly composed of two finite building blocks. If the two associated transfer matrices commute, the corresponding energy is called critical. Such critical energies appear in physical models, an example being the widely studied random dimer model. It is proven that the Lyapunov exponent vanishes quadratically at a generic critical energy and that the density of states is positive there. Large deviation estimates around these asymptotics allow to prove optimal lower bounds on quantum transport, showing that it is almost surely overdiffusive even though the models are known to have pure-point spectrum with exponentially localized eigenstates for almost every configuration of the polymers. Furthermore, the level spacing is shown to be regular at the critical energy.

Correlation Estimates in the Anderson Model

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 dis- tribution 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.

Operators with singular continuous spectrum, V. Sparse potentials

Abstract We give a new proof of correlation estimates for arbitrary moments of the resolvent of random Schrödinger operators on the lattice that generalizes and extends the correlation estimate of Minami for the second moment. We apply this moment bound to obtain a new n-level Wegner-type estimate that measures eigenvalue correlations through an upper bound on the probability that a local Hamiltonian has at least n eigenvalues in a given energy interval. Another consequence of the correlation estimates is that the results on the Poisson statistics of energy level spacing and the simplicity of the eigenvalues in the strong localization regime hold for a wide class of translation-invariant, selfadjoint, lattice operators with decaying off-diagonal terms and random potentials.

Dynamical Localization in Disordered Quantum Spin Systems

By presenting simple theorems for the absence of positive eigenvalues for certain one-dimensional Schrodinger operators, we are able to construct explicit potentials which yield purely singular continuous spectrum.

Expansions in generalized eigenfunctions of selfadjoint operators

We say that a quantum spin system is dynamically localized if the time-evolution of local observables satisfies a zero-velocity Lieb-Robinson bound. In terms of this definition we have the following main results: First, for general systems with short range interactions, dynamical localization implies exponential decay of ground state correlations, up to an explicit correction. Second, the dynamical localization of random xy spin chains can be reduced to dynamical localization of an effective one-particle Hamiltonian. In particular, the isotropic xy chain in random exterior magnetic field is dynamically localized.

Minimizing the Ground State Energy of an Electron in a Randomly Deformed Lattice

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