arXiv subjects

Peter Kramer

Publications and source records attributed to Peter Kramer.

34 records · Page 2Linked to original sources

arXiv2014

Interacting electrons in a magnetic field in a center-of-mass free basis

We present an extension of the spin-adapted configuration-interaction method for the computation of four electrons in a quasi two-dimensional quantum dot. By a group-theoretical decomposition of the basis set and working with relative and center-of-mass coordinates we obtain an analytical identification of all spurious center-of-mass states of the Coulomb-interacting electrons. We find a substantial reduction in the basis set used for numerical computations. At the same time we increase the accuracy compared to the standard spin-adapted configuration-interaction method (SACI) due to the absence of distortions caused by an unbalanced cut-off of center-of-mass excitations.

cond-mat.mes-hall↗

arXiv2012

Spherical Orbifolds for Cosmic Topology

Harmonic analysis is a tool to infer cosmic topology from the measured astrophysical cosmic microwave background CMB radiation. For overall positive curvature, Platonic spherical manifolds are candidates for this analysis. We combine the specific point symmetry of the Platonic manifolds with their deck transformations. This analysis in topology leads from manifolds to orbifolds. We discuss the deck transformations of the orbifolds and give eigenmodes for the harmonic analysis as linear combinations of Wigner polynomials on the 3-sphere. These provide new tools for detecting cosmic topology from the CMB radiation.

astro-ph.CO↗

arXiv2010

Gateways towards quasicrystals

The experimental discovery of quasicrystals by D Shechtman, D Gratias, I Blech, and J W Cahn in 1984 provided the paradigm for a new type of long-range order of solid matter in nature. This discovery stimulated an explosion of new experimental and theoretical research. In years prior to the discovery, there was a very active development of various gateways to quasicrystals in theoretical and mathematical physics. Without this conceptual basis, it would have been impossible to grasp and explore efficiently the structure and physical properties of quasicrystrals. The aim in what follows is to give a non-technical and condensed account of the conceptual gateways to quasicrystals prior to their discovery.

cond-mat.mtrl-sci↗

arXiv2010

Spherical spaces for cosmic topology and multipole selection rules

Spherical manifolds yield cosmic spaces with positive curvature. They result by closing pieces from the sphere used by Einstein for his initial cosmology. Harmonic analysis on the manifolds aims at explaining the observed low amplitudes at small multipole orders of the cosmic microwave background. We analyze assumptions of point symmetry and randomness for spherical spaces. There emerge four spaces named orbifolds, with low volume fraction from the sphere and sharp multipole selection rules in their eigenmodes.

astro-ph.CO↗

arXiv2010

Multipole analysis in cosmic topology

Low multipole amplitudes in the Cosmic Microwave Background CMB radiation can be explained by selection rules from the underlying multiply-connected homotopy. We apply a multipole analysis to the harmonic bases and introduce point symmetry.We give explicit results for two cubic 3-spherical manifolds and lowest polynomial degrees, and derive three new spherical 3-manifolds.

astro-ph.CO↗

arXiv2010

Platonic topology and CMB fluctuations: Homotopy, anisotropy, and multipole selection rules

The Cosmic Microwave Background CMB originates from an early stage in the history of the universe. Observed low multipole contributions of CMB fluctuations have motivated the search for selection rules from the underlying topology of 3-space. Everitt (2004) has generated all homotopies for Platonic spherical 3-manifolds by face gluings. We transform the glue generators into isomorphic deck transformations. The deck transformations act on a spherical Platonic 3-manifold as prototile and tile the 3-sphere by its images. A complete set of orthonormal functions on the 3-sphere is spanned by the Wigner harmonic polynomials. For a tetrahedral, two cubic and three octahedral manifolds we construct algebraically linear combinations of Wigner polynomials, invariant under deck transformations and with domain the manifold. We prove boundary conditions on polyhedral faces from homotopy. By algebraic means we pass to a multipole expansion. Assuming random models of the CMB radiation, we derive multipole selection rules, depending on the point symmetry of the manifold.

astro-ph.CO↗

arXiv2009

Platonic polyhedra tune the 3-sphere: III. Harmonic analysis on octahedral spherical 3-manifolds

From the homotopy groups of three distinct octahedral spherical 3-manifolds we construct the isomorphic groups H of deck transformations acting on the 3-sphere. The H-invariant polynomials on the 3-sphere constructed by representation theory span the bases for the harmonic analysis on three spherical manifolds. Analysis of the Cosmic Microwave Background in terms of these new bases can reveal a non-simple topology of the space part of space-time.

math-ph↗

arXiv2009

Platonic polyhedra tune the 3-sphere: II. Harmonic analysis on cubic spherical 3-manifolds

From the homotopy groups of two cubic spherical 3-manifolds we construct the isomorphic groups of deck transformations acting on the 3-sphere. These groups become the cyclic group of order eight and the quaternion group respectively. By reduction of representations from the orthogonal group to the identity representation of these subgroups we provide two subgroup-periodic bases for the harmonic analysis on the 3-manifolds. This harmonic analysis has applications to cosmic topology.

math.DG↗

arXiv2009

Platonic polyhedra tune the 3-sphere: Harmonic analysis on simplices

A spherical topological manifold of dimension n-1 forms a prototile on its cover, the (n-1)-sphere. The tiling is generated by the fixpoint-free action of the group of deck transformations. By a general theorem, this group is isomorphic to the first homotopy group. Multiplicity and selection rules appear in the form of reduction of group representations. A basis for the harmonic analysis on the (n-1)-sphere is given by the spherical harmonics which transform according to irreducible representations of the orthogonal group. The deck transformations form a subgroup, and so the representations of the orthogonal group can be reduced to those of this subgroup. Upon reducing to the identity representation of the subgroup, the reduced subset of spherical harmonics becomes periodic on the tiling and tunes the harmonic analysis on the (n-1)-sphere to the manifold. A particular class of spherical 3-manifolds arises from the Platonic polyhedra. The harmonic analysis on the Poincare dodecahedral 3-manifold was analyzed along these lines. For comparison we construct here the harmonic analysis on simplicial spherical manifolds of dimension n=1,2,3. Harmonic analysis applied to the cosmic microwave background by selection rules can provide evidence for multiply connected cosmic topologies.

math.DG↗

arXiv2008

Quasiperiodic propagation in time of some classical/quantum systems: Nielsen's conserved quantity and Floquet properties

We consider classical and quantum propagators for two different time intervals. If these propagators follow one another in a Fibonacci sequence we get a discrete quasiperiodic system. A theorem due to Nielsen provides a novel conserved quantity for this system. The Nielsen quantity controls the transition between commutative and non-commutative propagation in time. The quasiperiodically kicked oscillator moreover is dominated by quasiperiodic analogues of the Floquet theorem.

quant-ph↗

arXiv2007

Harmonic polynomials for expanding the fluctuations of the Cosmic Microwave Background: The Poincare and the 3-sphere model

Fluctuations of the Cosmic Microwave Background CMB are observed by the WMAP. When expanded into the harmonic eigenmodes of the space part of a cosmological model, they provide insight into the large-scale topology of space. All harmonic polynomials on the multiply connected dodecahedral Poincare space are constructed. Strong and specific selection rules are given by comparing the polynomials to those on the 3-sphere, its simply connected cover.

gr-qc↗

arXiv2005

Invariant operator due to F. Klein quantizes H. Poincare's dodecahedral 3-manifold

The eigenmodes of the Poincaré dodecahedral 3-manifold $M$ are constructed as eigenstates of a novel invariant operator. The topology of $M$ is characterized by the homotopy group $π_1(M)$, given by loop composition on $M$, and by the isomorphic group of deck transformations $deck(\tilde{M})$, acting on the universal cover $\tilde{M}$. ($π_1(M)$, $\tilde{M}$) are known to be the binary icosahedral group ${\cal H}_3$ and the sphere $S^3$ respectively. Taking $S^3$ as the group manifold $SU(2,C)$ it is shown that $deck(\tilde{M}) \sim {\cal H}^r_3$ acts on $SU(2,C)$ by right multiplication. A semidirect product group is constructed from ${\cal H}^r_3$ as normal subgroup and from a second group ${\cal H}^c_3$ which provides the icosahedral symmetries of $M$. Based on F. Klein's fundamental icosahedral ${\cal H}_3$-invariant, we construct a novel hermitian ${\cal H}_3$-invariant polynomial (generalized Casimir) operator ${\cal K}$. Its eigenstates with eigenvalues $κ$ quantize a complete orthogonal basis on Poincaré's dodecahedral 3-manifold. The eigenstates of lowest degree $λ=12$ are 12 partners of Klein's invariant polynomial. The analysis has applications in cosmic topology \cite{LA},\cite{LE}. If the Poincaré 3-manifold $M$ is assumed to model the space part of a cosmos, the observed temperature fluctuations of the cosmic microwave background must admit an expansion in eigenstates of ${\cal K}$.

gr-qc↗

arXiv2004

Group actions, geodesic loops, and symmetries of compact hyperbolic 3-manifolds

Compact hyperbolic 3-manifolds are used in cosmological models. Their topology is characterized by their homotopy group $π_1(M)$ whose elements multiply by path concatenation. The universal covering of the compact manifold $M$ is the hyperbolic space $H^3$ or the hyperbolic ball $B^3$. They share with $M$ a Riemannian metric of constant negative curvature and allow for the isometric action of the group $Sl(2,C)$. The homotopy group $π_1(M)$ acts as a uniform lattice $Γ(M)$ on $B^3$ and tesselates it by copies of $M$. Its elements $g$ produce preimage and image points for geodesic sections on $B^3$ which by self-intersection form geodesic loops on $M$. For any fixed hyperbolic $g \in Γ$ we construct a continuous commutative two-parameter normalizer $N_g <Sl(2,C)$ and its orbit surfaces on $B^3$. The orbit surfaces classify sets of geodesic loops of equal length. We give general expressions for the length of geodesic loops and for the defect angle at the self-intersection on $M$ in terms of the group parameters of $g$ and orbit parameters on $B^3$. Geodesic loops of minimal length, given from the character $χ(g)$, belong to a single orbit. These and only these minimal geodesic loops have vanishing defect angle and hence are smooth everywhere. The role of symmetries is illuminated by the example of the dodecahedral hyperbolic Weber-Seifert manifold $M$. $Γ(M)$ is normal in the hyperbolic Coxeter group with Coxeter diagram ${\bf \circ \frac{5}{}\circ\frac{3}{}\circ\frac{5}{}\circ}$. This leads to symmetry relations between geodesic loops.

astro-ph↗

arXiv1999

Quasicrystals: Atomic coverings and windows are dual projects

In the window approach to quasicrystals, the atomic position space E_parallel is embedded into a space E^n = E_parallel + E_perp. Windows are attached to points of a lattice Lambda \in E^n. For standard 5fold and icosahedral tiling models, the windows are perpendicular projections of dual Voronoi and Delone cells from Lambda. Their cuts by the position space E_parallel mark tiles and atomic positions. In the alternative covering approach, the position space is covered by overlapping copies of a quasi-unit cell which carries a fixed atomic configuration. The covering and window approach to quasicrystals are shown to be dual projects: D- and V- clusters are defined as projections to position space E_parallel of Delone or Voronoi cells. Decagonal V-clusters in the Penrose tiling, related to the decagon covering, and two types of pentagonal D-clusters in the triangle tiling of 5fold point symmetry with their windows are analyzed. They are linked, cover position space and have definite windows. For functions compatible with the tilings they form domains of definition. For icosahedral tilings the V-clusters are Kepler triacontahedra, the D-clusters are two icosahedra and one dodecahedron.

math-ph↗

arXiv1999

Exact electron states in 1D (quasi-) periodic arrays of delta-potentials

Exact one-electron eigenstates in finite parts of 1D periodic and Fibonacci chains of attractive and repulsive delta potentials are computed and analyzed. Bloch and bound state boundary conditions are related in terms of transfer matrices. Scenarios of positive and negative energy are distinguished. The dependence on the potential strength parameter is analyzed. The scattering matrix is computed. Implications for the interpretation of band germs in quasiperiodic chains are discussed.

math-ph↗

arXiv1999

Tiling theory applied to the surface structure of icosahedral AlPdMn quasicrystals

Surfaces in i-Al68Pd23Mn9 as observed with STM and LEED experiments show atomic terraces in a Fibonacci spacing. We analyze them in a bulk tiling model due to Elser which incorporates many experimental data. The model has dodecahedral Bergman clusters within an icosahedral tiling T^*(2F) and is projected from the 6D face-centered hypercubic lattice. We derive the occurrence and Fibonacci spacing of atomic planes perpendicular to any 5fold axis, compute the variation of planar atomic densities, and determine the (auto-) correlation functions. Upon interpreting the planes as terraces at the surface we find quantitative agreement with the STM experiments.

math-ph↗