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Simone Warzel

Publications and source records attributed to Simone Warzel.

87 records · Page 5Linked to original sources
arXiv2005

Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder

We consider the Laplacian on a rooted metric tree graph with branching number $ K \geq 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.

math-ph↗
arXiv2005

Ballistic transport in random magnetic fields with anisotropic long-ranged correlations

We present exact theoretical results about energetic and dynamic properties of a spinless charged quantum particle on the Euclidean plane subjected to a perpendicular random magnetic field of Gaussian type with non-zero mean. Our results refer to the simplifying but remarkably illuminating limiting case of an infinite correlation length along one direction and a finite but strictly positive correlation length along the perpendicular direction in the plane. They are therefore ``random analogs'' of results first obtained by A. Iwatsuka in 1985 and by J. E. Müller in 1992, which are greatly esteemed, in particular for providing a basic understanding of transport properties in certain quasi-two-dimensional semiconductor heterostructures subjected to non-random inhomogeneous magnetic fields.

cond-mat.dis-nn↗
arXiv2004

Anderson Localization and Lifshits Tails for Random Surface Potentials

We consider Schrödinger operators on $L^2(R^d)$ with a random potential concentrated near the surface $R^{d_1}\times\{0\}\subset 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] we infer Anderson localization (pure point spectrum and dynamical localization) for low energies. Our proof of Lifshits tail 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.

math-ph↗
arXiv2004

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: \mathbbm{R}^d \to [0, \infty[ $ decay at infinity in an anisotropic way, for example, $f(x_1,x_2)\sim (|x_1|^{α_1}+|x_2|^{α_2})^{-1}$ as $ |(x_1,x_2)| \to \infty $. As is expected from the isotropic situation, there is a so-called quantum regime with Lifshits exponent $ d/2 $ if both $α_1$ and $α_2$ are big enough, and there is a so-called classical regime with Lifshits exponent depending on $α_1$ and $α_2$ 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 $ α_1$ and $α_2$ simultaneously.

math-ph↗
arXiv2003

A survey of rigorous results on random Schroedinger operators for amorphous solids

Electronic properties of amorphous or non-crystalline disordered solids are often modelled by one-particle Schroedinger operators with random potentials which are ergodic with respect to the full group of Euclidean translations. We give a short, reasonably self-contained survey of rigorous results on such operators, where we allow for the presence of a constant magnetic field. We compile robust properties of the integrated density of states like its self-averaging, uniqueness and leading high-energy growth. Results on its leading low-energy fall-off, that is, on its Lifshits tail, are then discussed in case of Gaussian and non-negative Poissonian random potentials. In the Gaussian case with a continuous and non-negative covariance function we point out that the integrated density of states is locally Lipschitz continuous and present explicit upper bounds on its derivative, the density of states. Available results on Anderson localization concern the almost-sure pure-point nature of the low-energy spectrum in case of certain Gaussian random potentials for arbitrary space dimension. Moreover, under slightly stronger conditions all absolute spatial moments of an initially localized wave packet in the pure-point spectral subspace remain almost surely finite for all times. In case of one dimension and a Poissonian random potential with repulsive impurities of finite range, it is known that the whole energy spectrum is almost surely only pure point.

cond-mat.dis-nn↗
arXiv2003

Quantum-classical transitions in Lifshits tails with magnetic fields

We consider Lifshits' 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 Lifshits for short-ranged single-impurity potentials $ U $. The Lifshits tail is shown to change its character from purely quantum, through quantum-classical, to purely classical with 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.

cond-mat.dis-nn↗
arXiv2003

Quasi-classical versus non-classical spectral asymptotics for magnetic Schroedinger operators with decreasing electric potentials

We consider the Schroedinger operator H on L^2(R^2) or L^2(R^3) with constant magnetic field and electric potential V which typically decays at infinity exponentially fast or has a compact support. We investigate the asymptotic behaviour of the discrete spectrum of H near the boundary points of its essential spectrum. If the decay of V is Gaussian or faster, this behaviour is non-classical in the sense that it is not described by the quasi-classical formulas known for the case where V admits a power-like decay.

math-ph↗
arXiv2002

Simple diamagnetic monotonicities for Schroedinger operators with inhomogeneous magnetic fields of constant direction

Under certain simplifying conditions we detect monotonicity properties of the ground-state energy and the canonical-equilibrium density matrix of a spinless charged particle in the Euclidean plane subject to a perpendicular, possibly inhomogeneous magnetic field and an additional scalar potential. Firstly, we point out a simple condition warranting that the ground-state energy does not decrease when the magnetic field and/or the potential is increased pointwise. Secondly, we consider the case in which both the magnetic field and the potential are constant along one direction in the plane and give a genuine path-integral argument for corresponding monotonicities of the density-matrix diagonal and the absolute value of certain off-diagonals. Our results complement to some degree results of M. Loss and B. Thaller [Commun. Math. Phys. 186 (1997) 95] and L. Erdos [J. Math. Phys. 38 (1997) 1289].

quant-ph↗
arXiv2001

Existence and uniqueness of the integrated density of states for Schrödinger operators with magnetic fields and unbounded random potentials

The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schrödinger operator with a constant magnetic field and a random potential which may be unbounded from above and from below. For an ergodic random potential satisfying a simple moment condition, we give a detailed proof that the infinite-volume limits of spatial eigenvalue concentrations of finite-volume operators with different boundary conditions exist almost surely. Since all these limits are shown to coincide with the expectation of the trace of the spatially localized spectral family of the infinite-volume operator, the integrated density of states is almost surely non-random and independent of the chosen boundary condition. Our proof of the independence of the boundary condition builds on and generalizes certain results by S. Doi, A. Iwatsuka and T. Mine [Math. Z. {\bf 237} (2001) 335-371] and S. Nakamura [J. Funct. Anal. {\bf 173} (2001) 136-152].

math-ph↗
arXiv2001

Upper bounds on the density of states of single Landau levels broadened by Gaussian random potentials

We study a non-relativistic charged particle on the Euclidean plane R^2 subject to a perpendicular constant magnetic field and an R^2-homogeneous random potential in the approximation that the corresponding random Landau Hamiltonian on the Hilbert space L^2(R^2) is restricted to the eigenspace of a single but arbitrary Landau level. For a wide class of Gaussian random potentials we rigorously prove that the associated restricted integrated density of states is absolutely continuous with respect to the Lebesgue measure. We construct explicit upper bounds on the resulting derivative, the restricted density of states. As a consequence, any given energy is seen to be almost surely not an eigenvalue of the restricted random Landau Hamiltonian.

math-ph↗
arXiv2001

The absolute continuity of the integrated density of states for magnetic Schrödinger operators with certain unbounded random potentials

The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schr{ö}dinger operator with magnetic field and a random potential which may be unbounded from above and below. In case that the magnetic field is constant and the random potential is ergodic and admits a so-called one-parameter decomposition, we prove the absolute continuity of the integrated density of states and provide explicit upper bounds on its derivative, the density of states. This local Lipschitz continuity of the integrated density of states is derived by establishing a Wegner estimate for finite-volume Schrödinger operators which holds for rather general magnetic fields and different boundary conditions. Examples of random potentials to which the results apply are certain alloy-type and Gaussian random potentials. Besides we show a diamagnetic inequality for Schrödinger operators with Neumann boundary conditions.

math-ph↗
arXiv1999

The multiformity of Lifshits tails caused by random Landau Hamiltonians with repulsive impurity potentials of different decay at infinity

For a charged quantum particle in the Euclidean plane subject to a perpendicular constant magnetic field and repulsive impurities, randomly distributed according to Poisson's law, we determine the leading low-energy fall-off of the integrated density of states in case the single-impurity potential has either super-Gaussian or regular sub-Gaussian long-distance decay. The forms of the resulting so-called magnetic Lifshits tails reflect the great variety of these decays. On the whole, we summarize, unify, and generalize results in previous works of K. Broderix, L. Erdős, D.\ Hundertmark, W. Kirsch, and ourselves.

math-ph↗
arXiv1999

A rigorous path integral for quantum spin using flat-space Wiener regularization

Adapting ideas of Daubechies and Klauder [J. Math. Phys. {\bf 26} (1985) 2239] we derive a rigorous continuum path-integral formula for the semigroup generated by a spin Hamiltonian. More precisely, we use spin-coherent vectors parametrized by complex numbers to relate the coherent representation of this semigroup to a suitable Schrödinger semigroup on the Hilbert space $L^2(R^2)$ of Lebesgue square-integrable functions on the Euclidean plane $R^2$. The path-integral formula emerges from the standard Feynman-Kac-Itô formula for the Schrödinger semigroup in the ultra-diffusive limit of the underlying Brownian bridge on $R^2$. In a similar vein, a path-integral formula can be constructed for the coherent representation of the unitary time evolution generated by the spin Hamiltonian.

math-ph↗
arXiv1998

A rigorous path-integral formula for quantum-spin dynamics via planar Brownian motion

Adapting ideas of Daubechies and Klauder we derive a continuum path-integral formula for the time evolution generated by a spin Hamiltonian. For this purpose we identify the finite-dimensional spin Hilbert space with the ground-state eigenspace of a suitable Schödinger operator on $L^2({\mathbb{R}}^2)$, the Hilbert space of square-integrable functions on the Euclidean plane ${\mathbb{R}}^2$, and employ the Feynman-Kac-Itô formula.

quant-ph↗