Anderson localization and Lifshits tails for random surface potentials

We consider Schrödinger operators on L²(R^d) with a random potential concentrated near the surface R^d₁×{0}⊂R^d. We prove that the integrated density of states of such operators exhibits Lifshits tails near the bottom of the spectrum. From this and the multiscale analysis by Boutet de Monvel and Stollmann [Arch. Math. 80 (2003) 87-97] we infer Anderson localization (pure point spectrum and dynamical localization) for low energies. Our proof of Lifshits tails relies on spectral properties of Schrödinger operators with partially periodic potentials. In particular, we show that the lowest energy band of such operators is parabolic.     

Boosted Simon‐Wolff Spectral Criterion and Resonant Delocalization

Discussed here are criteria for the existence of continuous components in the spectra of operators with random potential. First, the essential condition for the Simon‐Wolff criterion is shown to be measurable at infinity. By implication, for the i.i.d. case and more generally potentials with the K‐property, the criterion is boosted by a zero‐one law. The boosted criterion, combined with tunneling estimates, is then applied for sufficiency conditions for the presence of continuous spectrum for random Schrödinger operators. The general proof strategy that this yields is modeled on the resonant delocalization arguments by which continuous spectrum in the presence of disorder was previously established for random operators on tree graphs. In another application of the Simon‐Wolff rank‐one analysis we prove the almost sure simplicity of the pure point spectrum for operators with random potentials of conditionally continuous distribution.© 2015 Wiley Periodicals, Inc.     

On Bernoulli decompositions for random variables, concentration bounds, and spectral localization

As was noted already by A. N. Kolmogorov, any random variable has a Bernoulli component. This observation provides a tool for the extension of results which are known for Bernoulli random variables to arbitrary distributions. Two applications are provided here: (i) an anti-concentration bound for a class of functions of independent random variables, where probabilistic bounds are extracted from combinatorial results, and (ii) a proof, based on the Bernoulli case, of spectral localization for random Schrödinger operators with arbitrary probability distributions for the single site coupling constants. For a general random variable, the Bernoulli component may be defined so that its conditional variance is uniformly positive. The natural maximization problem is an optimal transport question which is also addressed here.     

The Canopy Graph and Level Statistics for Random Operators on Trees

For operators with homogeneous disorder, it is generally expected that there is a relation between the spectral characteristics of a random operator in the infinite setup and the distribution of the energy gaps in its finite volume versions, in corresponding energy ranges. Whereas pure point spectrum of the infinite operator goes along with Poisson level statistics, it is expected that purely absolutely continuous spectrum would be associated with gap distributions resembling the corresponding random matrix ensemble. We prove that on regular rooted trees, which exhibit both spectral types, the eigenstate point process has always Poissonian limit. However, we also find that this does not contradict the picture described above if that is carefully interpreted, as the relevant limit of finite trees is not the infinite homogenous tree graph but rather a single-ended ‘canopy graph.’ For this tree graph, the random Schrödinger operator is proven here to have only pure-point spectrum at any strength of the disorder. For more general single-ended trees it is shown that the spectrum is always singular – pure point possibly with singular continuous component which is proven to occur in some cases.     

Energetic and Dynamic Properties of a Quantum Particle in a Spatially Random Magnetic Field with Constant Correlations along one Direction

We consider an electrically charged particle on the Euclidean plane subjected to a perpendicular magnetic field which depends only on one of the two Cartesian co-ordinates. For such a “unidirectionally constant” magnetic field (UMF), which otherwise may be random or not, we prove certain spectral and transport properties associated with the corresponding one-particle Schr?dinger operator (without scalar potential) by analysing its “energy-band structure”. In particular, for an ergodic random UMF we provide conditions which ensure that the operator’s entire spectrum is almost surely absolutely continuous. This implies that, along the direction in which the random UMF is constant, the quantum-mechanical motion is almost surely ballistic, while in the perpendicular direction in the plane one has dynamical localization. The conditions are verified, for example, for Gaussian and Poissonian random UMF’s with non-zero mean-values. These results may be viewed as “random analogues” of results first obtained by A. Iwatsuka [Publ. RIMS, Kyoto Univ. 21 (1985) 385] and (non-rigorously) by J.E. Müller [Phys. Rev. Lett. 68 (1992) 385].

Heinz BAUER (31 January 1928 - 15 August 2002)

of Erlangen-Nürnberg

Communicated by Frank den Hollander submitted 30/12/04, accepted 13/06/05     

Quantum-classical transitions in Lifshitz tails with magnetic fields

We consider Lifshitz's model of a quantum particle subject to a repulsive Poissonian random potential and address various issues related to the influence of a constant magnetic field on the leading low-energy tail of the integrated density of states. In particular, we propose the magnetic analog of a 40-year-old landmark result of Lifshitz for short-ranged single-impurity potentials U. The Lifshitz tail is shown to change its character from purely quantum, through quantum classical, to purely classical with an increasing range of U. This systematics is explained by the increasing importance of the classical fluctuations of the particle's potential energy in comparison to the quantum fluctuations associated with its kinetic energy.     

Lifshits Tails Caused by Anisotropic Decay: The Emergence of a Quantum-Classical Regime

We investigate Lifshits-tail behaviour of the integrated density of states for a wide class of Schrödinger operators with positive random potentials. The setting includes alloy-type and Poissonian random potentials. The considered (single-site) impurity potentials f: ?^d→[0,∞[ decay at infinity in an anisotropic way, for example, \(f(x_{1},x_{2})\sim (|x_{1}|^{\alpha_{1}}+|x_{2}|^{\alpha_{2}})^{-1}\) as |(x₁,x₂)|→∞. As is expected from the isotropic situation, there is a so-called quantum regime with Lifshits exponent d/2 if both α₁ and α₂ are big enough, and there is a so-called classical regime with Lifshits exponent depending on α₁ and α₂ if both are small. In addition to this we find two new regimes where the Lifshits exponent exhibits a mixture of quantum and classical behaviour. Moreover, the transition lines between these regimes depend in a nontrivial way on α₁ and α₂ simultaneously.     

Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder

We consider the Laplacian on a rooted metric tree graph with branching number K≥2 and random edge lengths given by independent and identically distributed bounded variables. Our main result is the stability of the absolutely continuous spectrum for weak disorder. A useful tool in the discussion is a function which expresses a directional transmission amplitude to infinity and forms a generalization of the Weyl-Titchmarsh function to trees. The proof of the main result rests on upper bounds on the range of fluctuations of this quantity in the limit of weak disorder.