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Werner Kirsch

Publications and source records attributed to Werner Kirsch.

At least 37 records · Page 2Linked to original sources
arXiv2019

Eigenvalue statistics for Schrödinger operators with random point interactions on $\mathbb{R}^d$, $d=1,2,3$

We prove that the local eigenvalue statistics at energy $E$ in the localization regime for Schrödinger operators with random point interactions on $\mathbb{R}^d$, for $d=1,2,3$, is a Poisson point process with the intensity measure given by the density of states at $E$ times the Lebesgue measure. This is one of the first examples of Poisson eigenvalue statistics for the localization regime of multi-dimensional random Schrödinger operators in the continuum. The special structure of resolvent of Schrödinger operators with point interactions facilitates the proof of the Minami estimate for these models.

math-ph↗
arXiv2018

Double Majority and Generalized Brexit: Explaining Counterintuitive Results

A mathematical analysis of the distribution of voting power in the Council of the European Union operating according to the Treaty of Lisbon is presented. We study the effects of Brexit on the voting power of the remaining members, measured by the Penrose--Banzhaf Index. We note that the effects in question are non-monotonic with respect to voting weights, and that some member states will lose power after Brexit. We use the normal approximation of the Penrose--Banzhaf Index in double-majority games to show that such non-monotonicity is in most cases inherent in the double-majority system, but is strongly exacerbated by the peculiarities of the EU population vector. Furthermore, we investigate consequences of a hypothetical "generalized Brexit", i.e., NN-exit of another member state (from a 28-member Union), noting that the effects on voting power are non-monotonic in most cases, but strongly depend on the size of the country leaving the Union.

physics.soc-ph↗
arXiv2018

Two Groups in a Curie-Weiss Model

We show for two disjoint groups of spins in a Curie-Weiss model and a homogeneous coupling matrix that the law of large numbers holds for the normed sums of both groups' spin variables. We also show that the central limit theorem holds only for low inverse temperatures.

math.PR↗
arXiv2018

An elementary proof of de Finetti's Theorem

A sequence of random variables is called exchangeable if the joint distribution of the sequence is unchanged by any permutation of the indices. De Finetti's theorem characterizes all $\{0,1\}$-valued exchangeable sequences as a "mixture" of sequences of independent random variables. We present an new, elementary proof of de Finetti's Theorem. The purpose of this paper is to make this theorem accessible to a broader community through an essentially self-contained proof.

math.PR↗
arXiv2018

Random matrices with exchangeable entries

We consider ensembles of real symmetric band matrices with entries drawn from an infinite sequence of exchangeable random variables, as far as the symmetry of the matrices permits. In general the entries of the upper triangular parts of these matrices are correlated and no smallness or sparseness of these correlations is assumed. It is shown that the eigenvalue distribution measures still converge to a semicircle but with random scaling. We also investigate the asymptotic behavior of the corresponding $\ell_2$-operator norms. The key to our analysis is a generalisation of a classic result by de Finetti that allows to represent the underlying probability spaces as averages of Wigner band ensembles with entries that are not necessarily centred. Some of our results appear to be new even for such Wigner band matrices.

math.PR↗
arXiv2018

Lifshits Tails for Squared Potentials

We consider Schrödinger operators with a random potential which is the square of an alloy-type potential. We investigate their integrated density of states and prove Lifshits tails. Our interest in this type of models is triggered by an investigation of randomly twisted waveguides.

math-ph↗
arXiv2018

Lifshits tails for randomly twisted quantum waveguides

We consider the Dirichlet Laplacian $H_γ$ on a 3D twisted waveguide with random Anderson-type twisting $γ$. We introduce the integrated density of states $N_γ$ for the operator $H_γ$, and investigate the Lifshits tails of $N_γ$, i.e. the asymptotic behavior of $N_γ(E)$ as $E \downarrow \inf {\rm supp}\, dN_γ$. In particular, we study the dependence of the Lifshits exponent on the decay rate of the single-site twisting at infinity.

math.SP↗
arXiv2017

Random Schrödinger operators with a background potential

We consider one-dimensional random Schrödinger operators with a background potential, arising in the inverse problem of scattering. We study the influence of the background potential on the essential spectrum of the random Schrödinger operator and obtain Anderson Localization for a larger class of one-dimensional Schrödinger operators. Further, we prove the existence of the integrated density of states and give a formula for it.

math-ph↗
arXiv2017

Semicircle law for generalized Curie-Weiss matrix ensembles at subcritical temperature

Hochstättler, Kirsch, and Warzel showed that the semicircle law holds for generalized Curie-Weiss matrix ensembles at or above the critical temperature. We extend their result to the case of subcritical temperatures for which the correlations between the matrix entries are stronger. Nevertheless, one may use the concept of approximately uncorrelated ensembles that was first introduced in the paper mentioned above. In order to do so one needs to remove the average magnetization of the entries by an appropriate modification of the ensemble that turns out to be of rank 1 thus not changing the limiting spectral measure.

math-ph↗
arXiv2017

Surface Lifshits tails for random quantum Hamiltonians

We consider Schrödinger operators on $L^{2}({\mathbb R}^{d})\otimes L^{2}({\mathbb R}^{\ell})$ of the form $ H_ω~=~H_{\perp}\otimes I_{\parallel} + I_{\perp} \otimes {H_\parallel} + V_ω$, where $H_{\perp}$ and $H_{\parallel}$ are Schrödinger operators on $L^{2}({\mathbb R}^{d})$ and $L^{2}({\mathbb R}^{\ell})$ respectively, and $ V_ω(x,y)$ : = $\sum_{ξ\in {\mathbb Z}^{d}} λ_ξ(ω) v(x - ξ, y)$, $x \in {\mathbb R}^d$, $y \in {\mathbb R}^\ell$, is a random 'surface potential'. We investigate the behavior of the integrated density of surface states of $H_ω$ near the bottom of the spectrum and near internal band edges. The main result of the current paper is that, under suitable assumptions, the behavior of the integrated density of surface states of $H_ω$ can be read off from the integrated density of states of a reduced Hamiltonian $H_{\perp}+W_ω$ where $W_ω$ is a quantum mechanical average of $V_ω$ with respect to $y \in {\mathbb R}^\ell$. We are particularly interested in cases when $H_{\perp}$ is a magnetic Schrödinger operator, but we also recover some of the results from [24] for non-magnetic $H_{\perp}$.

math-ph↗
arXiv2016

Sixty Years of Moments for Random Matrices

This is an elementary review, aimed at non-specialists, of results that have been obtained for the limiting distribution of eigenvalues and for the operator norms of real symmetric random matrices via the method of moments. This method goes back to a remarkable argument of Eugen Wigner some sixty years ago which works best for independent matrix entries, as far as symmetry permits, that are all centered and have the same variance. We then discuss variations of this classical result for ensembles for which the variance may depend on the distance of the matrix entry to the diagonal, including in particular the case of band random matrices, and/or for which the required independence of the matrix entries is replaced by some weaker condition. This includes results on ensembles with entries from Curie-Weiss random variables or from sequences of exchangeable random variables that have been obtained quite recently.

math-ph↗
arXiv2014

Semicircle law for a matrix ensemble with dependent entries

We study ensembles of random symmetric matrices whose entries exhibit certain correlations. Examples are distributions of Curie-Weiss-type. We provide a criterion on the correlations ensuring the validity of Wigner's semicircle law for the eigenvalue distribution measure. In case of Curie-Weiss distributions this criterion applies above the critical temperature (i.e. $β<1$). We also investigate the largest eigenvalue of certain ensembles of Curie-Weiss type and find a transition in its behavior at the critical temperature.

math-ph↗
arXiv2012

Lifshitz tails for a percolation model in the continuum

In this paper we study Lifshitz tails for continuous Laplacian in a continuous site percolation situation. By this we mean that we delete a random set $Γ_ω$ from $IR^d$ and consider the Dirichlet or Neumann Laplacian on $D=IR^d\setminusΓ_ω$. We prove that the integrated density of states exhibits Lifshitz behavior at the bottom of the spectrum when we consider Dirichlet boundary conditions, while when we consider Neumann boundary conditions, it is bounded from below by a van Hove behavior. The Lifshitz tails are proven independently of the percolation probability, whereas for the van Hove case we need some assumption on the volume of the sets taken out as well as on the percolation probability.

math-ph↗
arXiv2012

The fate of the square root law for correlated voting

We consider two-tier voting system and try to determine optimal weights for a fair representation in such systems. A prominent example of such a voting system is the Council of Ministers of the European Union. Under the assumption of independence of the voters, the square root law gives a fair distribution of power (based on the Penrose-Banzhaf power index) and a fair distribution of weights (based on the concept of the majority deficit), both given in the book by Felsenthal and Machover. In this paper, special emphasis is given to the case of correlated voters. The cooperative behaviour of the voters is modeled by suitable adoptions of spin systems known from statistical physics. Under certain assumptions we are able to compute the optimal weights as well as the average deviation of the council's vote from the public vote which we call the democracy deficit.

math.PR↗
arXiv2012

Transport in the random Kronig-Penney model

The Kronig-Penney model with random Dirac potentials on the lattice $\ZM$ has critical energies at which the Lyapunov exponent vanishes and the density of states has a van Hove singularity. This leads to a non-trivial quantum diffusion even though the spectrum is known to be pure-point.

math-ph↗
arXiv2010

Random Block Operators

We study fundamental spectral properties of random block operators that are common in the physical modelling of mesoscopic disordered systems such as dirty superconductors. Our results include ergodic properties, the location of the spectrum, existence and regularity of the integrated density of states, as well as Lifshits tails. Special attention is paid to the peculiarities arising from the block structure such as the occurrence of a robust gap in the middle of the spectrum. Without randomness in the off-diagonal blocks the density of states typically exhibits an inverse square-root singularity at the edges of the gap. In the presence of randomness we establish a Wegner estimate that is valid at all energies. It implies that the singularities are smeared out by randomness, and the density of states is bounded. We also show Lifshits tails at these band edges. Technically, one has to cope with a non-monotone dependence on the random couplings.

math-ph↗
arXiv2010

Lifshitz tails for a class of Schrödinger operators with random breather-type potential

We derive bounds on the integrated density of states for a class of Schrödinger operators with a random potential. The potential depends on a sequence of random variables, not necessarily in a linear way. An example of such a random Schrödinger operator is the breather model, as introduced by Combes, Hislop and Mourre. For these models we show that the integrated density of states near the bottom of the spectrum behaves according to the so called Lifshitz asymptotics. This result can be used to prove Anderson localization in certain energy/disorder regimes.

math-ph↗