arXiv subjects

Peter Müller

Publications and source records attributed to Peter Müller.

131 records · Page 8Linked to original sources

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

Finiteness Results for Hilbert's Irreducibility Theorem

Let f(t,X) be an irreducible polynomial over the field of rational functions k(t), where k is a number field. Let O be the ring of integers of k. Hilbert's irreducibility theorem gives infinitely many integral specializations of t to values a in O such that f(a,X) is still irreducible. In this paper we study the set Red(O) of those a in O with f(a,X) reducible. We show that Red(O) is a finite set under rather weak assumptions. In particular, several results of K. Langmann, obtained by Diophantine approximation techniques, appear as special cases of some of our results. Our method is completely different. We use elementary group theory, valuation theory, and Siegel's theorem about integral points on algebraic curves. Indeed, using the Siegel-Lang extension of Siegel's theorem, most of our results hold over more general fields.

math.NT↗

arXiv2001

Anomalous stress relaxation in random macromolecular networks

Within the framework of a simple Rouse-type model we present exact analytical results for dynamical critical behaviour on the sol side of the gelation transition. The stress-relaxation function is shown to exhibit a stretched-exponential long-time decay. The divergence of the static shear viscosity is governed by the critical exponent $k=ϕ-β$, where $ϕ$ is the (first) crossover exponent of random resistor networks, and $β$ is the critical exponent for the gel fraction. We also derive new results on the behaviour of normal stress coefficients.

cond-mat.stat-mech↗

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

Dynamical properties of random Schrödinger operators

We study dynamical properties of random Schrödinger operators $H^{(ω)}$ defined on the Hilbert space $\ell^2(\bbZ^d)$ or $L^2(\bbR^d)$. Building on results from existing multi-scale analyses, we give sufficient conditions on $H^{(ω)}$ to obtain the vanishing of the diffusion exponent $$ σ_{\rm diff}^+ := \limsup_{T\rightarrow\infty } \frac{\log \bbE \left(\la\la\vert X \vert^2\ra\ra_{T,f_I(H^{(ω)})ψ}\right) }{\log T}=0. $$ Here $\bbE$ is the expectation over randomness, $f_{I}$ is any smooth characteristic function of a bounded energy-interval $I$ and $ψ$ is a state vector in the domain of $H^{(ω)}$ with compact spatial support. The quantity $\la\la |X|^2 \ra\ra_{T,φ}$ denotes the Cesaro mean up to time $T$ of the second moment of position $\la |X|^2\ra_{t,φ}$ at times $0\le t\le T$ of an initial state vector $φ$. If the Hilbert space is $\ell^2(\bbZ^d)$, the method of proof can be strengthened to yield dynamical localization. Under weaker assumptions, we also prove a theorem on the absence of diffusion. The results are applied to a randomly perturbed periodic Schrödinger operator on $L^2(\bbR^d)$, to a simple Anderson-type model on the lattice and to a model with a correlated random potential in continuous space.

math-ph↗