% Encoding: UTF-8
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@article{faucris.120538484,
abstract = {We analyze the Farey spin chain, a one-dimensional spin system with long-range multibody interactions. Using a polymer model technique, we show that when the temperature is decreased below the (single) critical temperature Tc = 1/2, the magnetization jumps from zero to one.},
author = {Contucci, Pierluigi and Kleban, Peter and Knauf, Andreas},
doi = {10.1023/A:1004607107241},
faupublication = {no},
journal = {Journal of Statistical Physics},
keywords = {Magnetization; Spin chain},
note = {UnivIS-Import:2015-03-05:Pub.1999.nat.dma.lma6.afully},
pages = {523-539},
peerreviewed = {Yes},
title = {{A} {Fully} {Magnetizing} {Phase} {Transition}},
volume = {97},
year = {1999}
}
@article{faucris.210078916,
abstract = {Asymptotic
velocity is defined as the Cesàro limit of velocity. As such, its
existence has been proved for bounded interaction potentials. This is
known to be wrong in celestial mechanics with four or more bodies. Here,
we show for a class of pair potentials including the homogeneous ones
of degree −*α* for *α*∈(0, 2), that asymptotic velocities exist for up to four bodies, dimension three or larger, for any energy and *almost all* initial conditions on the energy surface.

This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

},
author = {Knauf, Andreas},
doi = {10.1098/rsta.2017.0426},
faupublication = {yes},
journal = {Philosophical Transactions of the Royal Society A-Mathematical Physical and Engineering Sciences},
keywords = {Celestial mechanics; Asymptotic velocity; Scattering theory},
peerreviewed = {Yes},
title = {{Asymptotic} velocity for four celestial bodies},
volume = {376},
year = {2018}
}
@incollection{faucris.117565404,
abstract = {
The classical motion of a particle in an attracting Coulomb (-1/*r*) potential is described by conic sections, whereas the equations of motion in the field of two fixed centres have been solved by Euler and Jacobi.

},
author = {Klein, Markus and Knauf, Andreas},
booktitle = {Mathematical physics, X (Leipzig, 1991)},
doi = {10.1007/978-3-642-77303-7{\_}27},
editor = {Konrad Schmüdgen},
faupublication = {no},
pages = {308--312},
peerreviewed = {unknown},
publisher = {Springer, Berlin},
title = {{Chaotic} motion in {Coulombic} potentials},
year = {1992}
}
@article{faucris.122426524,
abstract = {We consider the motion of a classical particle under the influence of a random potential on R^d, in particular the distribution of asymptotic velocities and the question of ergodicity of time evolution.},
author = {Knauf, Andreas and Schumacher, Christoph},
doi = {10.1017/etds.2012.141},
faupublication = {yes},
journal = {Ergodic Theory and Dynamical Systems},
pages = {557--593},
peerreviewed = {Yes},
title = {{Classical} motion in random potentials},
volume = {34},
year = {2014}
}
@article{faucris.265534746,
abstract = {We consider the scattering of n classical particles interacting via pair potentials which are assumed to be 'long-range', i.e. of order O(r(-alpha)) r tends to infinity, for some alpha > 0. We define and focus on the 'free region', the set of states leading to well-defined and well-separated final states at infinity. As a first step, we prove the existence of an explicit, global surface of section for the free region. This surface of section allows us to prove the smoothness of the map sending a point to its final state and to establish a forward conjugacy between the n-body dynamics and free dynamics.},
author = {Fejoz, Jacques and Knauf, Andreas and Montgomery, Richard},
doi = {10.1088/1361-6544/ac288d},
faupublication = {yes},
journal = {Nonlinearity},
note = {CRIS-Team WoS Importer:2021-10-29},
pages = {8017-8054},
peerreviewed = {Yes},
title = {{Classical} n-body scattering with long-range potentials},
volume = {34},
year = {2021}
}
@book{faucris.117563644,
author = {Sinai, Yakov G. and Knauf, Andreas},
doi = {10.1007/978-3-0348-8932-2},
faupublication = {no},
isbn = {3-7643-5708-8},
pages = {vi+98},
peerreviewed = {unknown},
publisher = {Birkhäuser Verlag, Basel},
series = {DMV Seminar},
title = {{Classical} nonintegrability, quantum chaos. {With} a contribution by {Viviane} {Baladi}},
volume = {27},
year = {1997}
}
@book{faucris.124012284,
abstract = {Astronomy as well as molecular physics describe non-relativistic motion by an interaction of the same form: By Newton's respectively by Coulomb's potential. But whereas the fundamental laws of motion thus have a simple form, the n-body problem withstood (for n > 2) all attempts of an explicit solution. Indeed, the studies of Poincare at the end of the last century lead to the conclusion that such an explicit solution should be impossible. Poincare himselfopened a new epoch for rational mechanics by asking qual itative questions like the one about the stability of the solar system. To a largeextent, his work, which was critical for the formation of differential geometry and topology, was motivated by problems arising in the analysis of the n-body problem ([38], p. 183). As it turned out, even by confining oneselfto questions ofqualitativenature, the general n-body problem could not be solved. Rather, simplified models were treated, like planar motion or the restricted 3-body problem, where the motion of a test particle did not influence the other two bodies.},
address = {Berlin, Heidelberg, New York},
author = {Klein, Markus and Knauf, Andreas},
doi = {10.1007/978-3-540-47336-7},
faupublication = {no},
note = {UnivIS-Import:2015-04-02:Pub.1993.nat.dma.lma6.classi},
publisher = {Springer},
series = {Lecture Notes in Physics},
title = {{Classical} {Planar} {Scattering} by {Coulombic} {Potentials}},
volume = {13},
year = {1993}
}
@article{faucris.117566724,
abstract = {The author considers the motion of a classical particle in a potential for values of the energy such that the particle cannot escape to infinity. It is known that in that case there exists at least one closed trajectory. Then for more than one freedom and energies smaller than the maximum of the potential the energy shell is not foliated by invariant tori. If any energy shell of a particle moving on a torus is foliated by invariant tori, then the potential is constant.},
author = {Knauf, Andreas},
doi = {10.1088/0951-7715/3/3/019},
faupublication = {no},
journal = {Nonlinearity},
pages = {961--973},
peerreviewed = {Yes},
title = {{Closed} orbits and converse {KAM} theory},
url = {http://stacks.iop.org/0951-7715/3/961},
volume = {3},
year = {1990}
}
@article{faucris.117616224,
abstract = {We study the propagation of phase space singularities for the time dependent Schrödinger equation with potential having Coulomb-type singularities in space dimension equal to*three*. We prove that the singularities (frequency set) of the solution are reflected by a Coulomb center exactly as in the classical problem, i.e. the frequency set follows the*regularized trajectories* of Classical Mechanics after a collision.},
author = {Gérard, Claude and Knauf, Andreas},
doi = {10.1007/BF02100283},
faupublication = {no},
journal = {Communications in Mathematical Physics},
pages = {17-26},
peerreviewed = {Yes},
title = {{Collisions} for the quantum {Coulomb} {Hamiltonian}},
volume = {143},
year = {1991}
}
@article{faucris.117567164,
abstract = {The motion of a quantum mechanical particle in a 2-dimensional crystal with attracting nuclei is considered. For a large class of potentials the particle is shown to be slow not only for low but also for *high* energies. The remarkable high energy behaviour is a manifestation of “quantum chaos.”},
author = {Knauf, Andreas},
doi = {10.1016/0003-4916(89)90315-1},
faupublication = {no},
journal = {Annals of Physics},
pages = {205--240},
peerreviewed = {Yes},
title = {{Coulombic} periodic potentials: {The} {Quantum} {Case}},
volume = {191},
year = {1989}
}
@incollection{faucris.118916424,
abstract = {
The usual operational definition of the term ‘Quantum Chaos’, meaning a quantum system whose classical counterpart is non-integrable, is not self-contained. However it is argued that there cannot be any intrinsic definition of chaoticity of a finite quantum system which is not based on some kind of semiclassical limit.

Unlike for finite systems, the quantum dynamical entropy of infinite systems may be strictly positive. However, an example shows that this quantity may be lowered by interactions which lead to an increase of classical dynamical entropy.

},
address = {Berlin, Heidelberg},
author = {Knauf, Andreas},
booktitle = {Chance in Physics},
doi = {10.1007/3-540-44966-3{\_}17},
editor = {Jean Bricmont, Giancarlo Ghirardi, Detlef Dürr, Francesco Petruccione, Maria Carla Galavotti, Nino Zanghi},
faupublication = {yes},
note = {UnivIS-Import:2015-04-20:Pub.2001.nat.dma.lma6.doesqu},
pages = {235-241},
peerreviewed = {unknown},
publisher = {Springer},
series = {Lecture Notes in Physics},
title = {{Does} {Quantum} {Chaos} {Exist}?},
volume = {547},
year = {2001}
}
@article{faucris.117559464,
abstract = {We study a curve of Gibbsian families of complex 3 × 3-matrices and point out new features, absent in commutative finite-dimensional algebras: a discontinuous maximum-entropy inference, a discontinuous entropy distance, and non-exposed faces of the mean value set. We analyze these problems from various aspects including convex geometry, topology, and information geometry. This research is motivated by a theory of infomax principles, where we contribute by computing first order optimality conditions of the entropy distance.},
author = {Knauf, Andreas and Weis, Stephan},
doi = {10.1063/1.4757652},
faupublication = {yes},
journal = {Journal of Mathematical Physics},
pages = {102206, 25},
peerreviewed = {Yes},
title = {{Entropy} distance: {New} quantum phenomena},
volume = {53},
year = {2012}
}
@article{faucris.117569144,
abstract = {The motion of a classical pointlike particle in a two-dimensional periodic potential with negative coulombic singularities is examined. This motion is shown to be Bernoullian for many potentials and high enough energies. Then the motion on the plane is a diffusion process. All such motions are topologically conjugate and the periodic orbits can be analysed with the help of a group.},
author = {Knauf, Andreas},
doi = {10.1007/BF01209018},
faupublication = {no},
journal = {Communications in Mathematical Physics},
pages = {89--112},
peerreviewed = {Yes},
title = {{Ergodic} and topological properties of {Coulombic} periodic potentials},
url = {http://projecteuclid.org/euclid.cmp/1104159170},
volume = {110},
year = {1987}
}
@article{faucris.106651644,
author = {Knauf, Andreas},
doi = {10.1007/s002200050715},
faupublication = {no},
journal = {Communications in Mathematical Physics},
pages = {491},
peerreviewed = {Yes},
title = {{Erratum}: ``{The} number-theoretical spin chain and the {Riemann} zeros'' [{Comm}.\ {Math}.\ {Phys}. 6 (1998), no. 3, 703--731; {MR}1645212 (2000d:11108)]},
volume = {206},
year = {1999}
}
@article{faucris.117561444,
author = {Guerra, Francesco and Knauf, Andreas},
doi = {10.1063/1.532247},
faupublication = {no},
journal = {Journal of Mathematical Physics},
pages = {3188--3202},
peerreviewed = {Yes},
title = {{Free} energy and correlations of the number-theoretical spin chain},
volume = {39},
year = {1998}
}
@article{faucris.117559904,
abstract = {We consider a family of Markov maps on the unit interval, interpolating between the tent map and the Farey map. The latter is not uniformly expanding. Each map being composed of two fractional linear transformations, the family generalizes many particular properties which for the case of the Farey map have been successfully exploited in number theory. We analyze the dynamics through the spectral properties of the generalized transfer operator. Application of the thermodynamic formalism to the family reveals first and second order phase transitions and unusual properties like positivity of the interaction function.},
author = {Knauf, Andreas and Degli Esposti, Mirko and Isola, Stefano},
doi = {10.1007/s00220-007-0294-3},
faupublication = {yes},
journal = {Communications in Mathematical Physics},
pages = {297--329},
peerreviewed = {Yes},
title = {{Generalized} {Farey} trees, transfer operators and phase transitions},
volume = {275},
year = {2007}
}
@article{faucris.239471633,
abstract = {We show that molecular precursors, combining six given segment types, can produce all carbon nanotubes of chiralities larger than (9,0).},
author = {Amsharov, Konstantin and Knauf, Andreas and Tomada, Jörg},
doi = {10.1016/j.dam.2020.05.031},
faupublication = {yes},
journal = {Discrete Applied Mathematics},
keywords = {Carbon nanotubes; Chirality; Rational synthesis},
note = {CRIS-Team Scopus Importer:2020-06-23},
pages = {55-60},
peerreviewed = {Yes},
title = {{Generating} carbon nanotube caps of almost all chiralities, using six molecular segment types},
volume = {285},
year = {2020}
}
@incollection{faucris.122221704,
author = {Golin, Simon and Knauf, Andreas and Marmi, Stefano},
booktitle = {Nonlinear dynamics (Bologna, 1988)},
faupublication = {no},
pages = {200--209},
peerreviewed = {unknown},
publisher = {World Sci. Publ., Teaneck, NJ},
title = {{Hannay} angles: existence and geometrical interpretation},
year = {1989}
}
@article{faucris.222560826,
author = {Fleischer, Stefan and Knauf, Andreas},
doi = {10.1007/s00205-019-01406-4},
faupublication = {yes},
journal = {Archive for Rational Mechanics and Analysis},
peerreviewed = {Yes},
title = {{Improbability} of {Collisions} in n-{Body} {Systems}.},
year = {2019}
}
@article{faucris.210079502,
abstract = {Given a volume preserving dynamical system with non-compact phase space,
one is sometimes interested in special subsets of its wandering set.
One example from celestial mechanics is the set of initial values
leading to collision. Another one is the set of initial values of
semi-orbits, whose asymptotic velocity does not exist as a limit. We
introduce techniques that can be helpful in showing that these sets are
of measure zero. We do this by defining a sequence of hypersurfaces,
that are eventually hit by each of those semi-orbits and whose total
surface area decreases to zer},
author = {Fleischer, Stefan and Knauf, Andreas},
doi = {10.1007/s00205-018-1309-2},
faupublication = {yes},
journal = {Archive for Rational Mechanics and Analysis},
pages = {1781–1800},
peerreviewed = {Yes},
title = {{Improbability} of {Wandering} {Orbits} {Passing} {Through} a {Sequence} of {Poincaré} {Surfaces} of {Decreasing} {Size}},
volume = {231},
year = {2018}
}
@article{faucris.122869384,
abstract = {It is shown that for generic configuration of the centres at high energy levels the n-centre problem is completely integrable by using $C^\infty$ integrals of the motion however it is not integrable in terms of real analytic functions},
author = {Knauf, Andreas and Taimanov, Iskander A.},
faupublication = {yes},
journal = {Doklady Mathematics},
peerreviewed = {Yes},
title = {{Integrability} of the n-center problem at high energies},
volume = {70},
year = {2004}
}
@incollection{faucris.120924804,
abstract = {Isaac Newton had one insight that he considered to be so fundamental that he kept it secret: “It is useful to solve differential equations.”^{1}
The notion of dynamical systems grew out of the theory of differential equations. It was realized by Henri Poincaré that equations like the one of the celestial three body problem could not be solved analytically. Thus it was necessary to supplement the quantitative approximate solutions by qualitative methods in order to understand the long time behaviour of the solutions of differential equations.},
address = {Heidelberg},
author = {Knauf, Andreas},
booktitle = {The Mathematical Aspects of Quantum Maps},
doi = {10.1007/3-540-37045-5{\_}1},
editor = {Mirko Degli Esposti, Sandro Graffi},
faupublication = {yes},
note = {UnivIS-Import:2015-04-02:Pub.2001.nat.dma.lma6.introd},
pages = {1-24},
peerreviewed = {unknown},
publisher = {Springer},
series = {Lecture Notes in Physics},
title = {{Introduction} to {Dynamical} {Systems}},
volume = {618},
year = {2003}
}
@incollection{faucris.115087324,
address = {Berlin, New York},
author = {Knauf, Andreas},
booktitle = {Geometrie und Physik},
editor = {Seiler Ruedi, Enss Volker, Müller Werner},
faupublication = {no},
isbn = {978-3-11-013944-0},
peerreviewed = {unknown},
publisher = {De Gruyter},
series = {Akademie der Wissenschaften zu Berlin, Forschungsberichte},
title = {{Inverse} {KAM}-{Theorie}},
volume = {8},
year = {1993}
}
@incollection{faucris.117564524,
abstract = {In this paper we present results on the relations between wave scattering in the so-called modular domain, analytic number theory, and classical statistical mechanics. The motion in the modular domain is an example of so-called arithmetic chaos. In this paper we shortly review results from [5, 6, 7], but concentrate on previously unpublished results.},
author = {Knauf, Andreas},
booktitle = {Stochasticity and quantum chaos (Sobótka Castle, 1993)},
doi = {10.1007/978-94-011-0169-1{\_}13},
editor = {Wojciech Cegła, Zbigniew Haba, Lech Jakóbczyk},
faupublication = {no},
pages = {137--148},
peerreviewed = {unknown},
publisher = {Kluwer Acad. Publ., Dordrecht},
series = {Mathematics and its Applications},
title = {{Irregular} scattering, number theory, and statistical mechanics},
volume = {317},
year = {1995}
}
@article{faucris.210079783,
abstract = {Motivated by the high-energy limit of the N-body problem we construct non-deterministic billiard process. The billiard table is the complement of a finite collection of linear subspaces within a Euclidean vector space. A trajectory is a constant speed polygonal curve with vertices on the subspaces and change of direction upon hitting a subspace governed by 'conservation of momentum' (mirror reflection). The itinerary of a trajectory is the list of subspaces it hits, in order. (A) Are itineraries finite? (B) What is the structure of the space of all trajectories having a fixed itinerary? In a beautiful series of papers Burago-Ferleger-Kononenko [BFK] answered (A) affirmatively by using non-smooth metric geometry ideas and the notion of a Hadamard space. We answer (B) by proving that this space of trajectories is diffeomorphic to a Lagrangian relation on the space of lines in the Euclidean space. Our methods combine those of BFK with the notion of a generating family for a Lagrangian relation.},
author = {Féjoz, Jacques and Knauf, Andreas and Montgomery, Richard},
doi = {10.1088/1361-6544/aa5b26},
faupublication = {yes},
journal = {Nonlinearity},
keywords = {billiards;N-body;Lagrangian Relations;CAT(0);Hadamard space;generating families},
pages = {1326-1355},
peerreviewed = {Yes},
title = {{Lagrangian} relations and linear point billiards},
volume = {30},
year = {2017}
}
@book{faucris.121317944,
abstract = {Als Grenztheorie der Quantenmechanik besitzt die klassische Dynamik einen großen Formenreichtum – vom gut berechenbaren bis zum chaotischen Verhalten. Ausgehend von interessanten Beispielen wird in dem Band nicht nur eine gelungene Auswahl grundlegender Themen vermittelt, sondern auch der Einstieg in viele aktuelle Forschungsgebiete im Bereich der klassischen Mechanik. Didaktisch geschickt aufgebaut und mit hilfreichen Anhängen versehen, werden lediglich Kenntnisse der Grundvorlesungen in Mathematik vorausgesetzt},
author = {Knauf, Andreas},
faupublication = {yes},
isbn = {978-3642209772},
peerreviewed = {unknown},
publisher = {Springer},
series = {Springer Masterclass},
title = {{Mathematische} {Physik}: {Klassische} {Mechanik}},
year = {2011}
}
@article{faucris.124159904,
abstract = {Stochastic interdependence of a probability distribution on a product space is measured by its Kullback–Leibler distance from the exponential family of product distributions (called multi-information). Here we investigate low-dimensional exponential families that contain the maximizers of stochastic interdependence in their closure. Based on a detailed description of the structure of probability distributions with globally maximal multi-information we obtain our main result: The exponential family of pure pair-interactions contains all global maximizers of the multi-information in its closure.},
author = {Knauf, Andreas and Ay, Nihat},
faupublication = {yes},
journal = {Kybernetika},
pages = {517--538},
peerreviewed = {Yes},
title = {{Maximizing} {Multi}-information},
volume = {42},
year = {2006}
}
@article{faucris.106645484,
abstract = {

We study many-party correlations quantified in terms of the Umegaki relative entropy (divergence) from a Gibbs family known as a hierarchical model. We derive these quantities from the maximum-entropy principle which was used earlier to define the closely related irreducible correlation. We point out the differences between quantum states and probability vectors which exist in hierarchical models, in the divergence from a hierarchical model and in local maximizers of this divergence. The differences are, respectively, missing factorization, discontinuity and reduction of uncertainty. We discuss global maximizers of the mutual information of separable qubit states.

},
author = {Knauf, Andreas and Weis, Stephan and Ay, Nihat and Zhao, Ming-Jing},
doi = {10.1142/S1230161215500067},
faupublication = {yes},
journal = {Open Systems & Information Dynamics},
pages = {1550006, 22},
peerreviewed = {Yes},
title = {{Maximizing} the divergence from a hierarchical model of quantum states},
volume = {22},
year = {2015}
}
@article{faucris.310449928,
author = {Knauf, Andreas},
doi = {10.1063/5.0153118},
faupublication = {yes},
journal = {Journal of Mathematical Physics},
note = {CRIS-Team Scopus Importer:2023-09-15},
peerreviewed = {Yes},
title = {{Mini}-course: {Classical} mechanics and transport},
volume = {64},
year = {2023}
}
@article{faucris.117562324,
author = {Asch, Joachim and Knauf, Andreas},
doi = {10.1088/0951-7715/11/1/011},
faupublication = {no},
journal = {Nonlinearity},
pages = {175--200},
peerreviewed = {Yes},
title = {{Motion} in periodic potentials},
volume = {11},
year = {1998}
}
@inproceedings{faucris.117617324,
abstract = {This paper presents a new derivation of a nonseparable multiresolution inversion formula in 3D Feldkamp-type cone-beam tomography. The approximative inverse is used to derive an inversion formula for reconstructing the wavelet approximation and wavelet detail coefficients of the volume slice by slice. Beyond the reconstruction algorithm, applications of multiresolution reconstruction in the field of non-destructive testing are shown: The algorithm supports progressive reconstruction and local tomography for recovering only a region of interest inside the investigated volume. The features of the reconstruction algorithm are shown by means of simulated data as well as measured data of an aluminium casting from automobile industry.},
author = {Knauf, Andreas and Oeckl, Steven and Schön, Tobias and Louis, Alfred K.},
booktitle = {Proceedings of the 9th European Conference on Non-Destructive Testing (ECNDT)},
faupublication = {yes},
isbn = {3-931381-86-2},
peerreviewed = {unknown},
title = {{Multiresolution} {3D}-{Computerized} {Tomography} and its {Application} to {NDT}},
venue = {Berlin},
year = {2006}
}
@article{faucris.115618184,
abstract = {In these lecture notes connections between the Riemann zeta function, motion in the modular domain and systems of statistical mechanics are presented.},
author = {Knauf, Andreas},
faupublication = {yes},
journal = {Reviews in Mathematical Physics},
note = {UnivIS-Import:2015-03-05:Pub.1999.nat.dma.lma6.number},
pages = {1027-1060},
peerreviewed = {Yes},
title = {{Number} {Theory}, {Dynamical} {Systems} and {Statistical} {Mechanics}},
volume = {11},
year = {1999}
}
@article{faucris.113821884,
abstract = {The quotient ξ(*s*-1)/ξ(*s*) of Riemann zeta functions is shown to be the partition function of a ferromagnetic spin chain for inverse temperature*s*.},
author = {Knauf, Andreas},
doi = {10.1007/BF02099041},
faupublication = {no},
journal = {Communications in Mathematical Physics},
pages = {77--115},
peerreviewed = {Yes},
title = {{On} a ferromagnetic spin chain},
url = {http://projecteuclid.org/euclid.cmp/1104252597},
volume = {153},
year = {1993}
}
@article{faucris.117565184,
abstract = {The existence of the thermodynamic limit of the free energy for the ferromagnetic spin chain connected with the Riemann zeta function is proven.},
author = {Knauf, Andreas},
doi = {10.1063/1.530769},
faupublication = {no},
journal = {Journal of Mathematical Physics},
pages = {228--236},
peerreviewed = {Yes},
title = {{On} a ferromagnetic spin chain. {II}. {Thermodynamic} limit},
volume = {35},
year = {1994}
}
@article{faucris.123533344,
abstract = {The soft billiard system is shown to have non-zero Lyapunov exponent except for those parameter values where zero exponents were already known to occur.},
author = {Knauf, Andreas},
doi = {10.1016/0167-2789(89)90084-5},
faupublication = {no},
journal = {Physica D-Nonlinear Phenomena},
pages = {259-262},
peerreviewed = {Yes},
title = {{On} {Soft} {Billiard} {Systems}},
volume = {36},
year = {1989}
}
@article{faucris.117566284,
author = {Helffer, Bernard and Knauf, Andreas and Siedentop, Heinz and Weikard, Rudi},
doi = {10.1080/03605309208820856},
faupublication = {no},
journal = {Communications in Partial Differential Equations},
pages = {615--639},
peerreviewed = {Yes},
title = {{On} the absence of a first order correction for the number of bound states of a {Schrödinger} operator with {Coulomb} singularity},
volume = {17},
year = {1992}
}
@article{faucris.123235684,
abstract = {We study the combinatorics of the contributions to the form factor of the group U(N) in the large N limit. This relates to questions about semiclassical contributions to the form factor of quantum systems described by the unitary ensemble.},
author = {Knauf, Andreas and Degli Esposti, Mirko},
doi = {10.1063/1.1814419},
faupublication = {yes},
journal = {Journal of Mathematical Physics},
pages = {4956--4979},
peerreviewed = {Yes},
title = {{On} the {Form} {Factor} for the {Unitary} {Group}},
volume = {45},
year = {2004}
}
@article{faucris.117560344,
abstract = {It is known that for *n*≥3 centres and positive energies the *n*-centre problem of celestial mechanics leads to a flow with a strange repellor and positive topological entropy. Here we consider the energies above some threshold and show: Whereas for arbitrary *g*>1 independent integrals of Gevrey class *g* exist, no real-analytic (that is, Gevrey class 1) independent integral exists.},
author = {Knauf, Andreas and Taimanov, Iskander A.},
doi = {10.1007/s00208-004-0598-y},
faupublication = {yes},
journal = {Mathematische Annalen},
pages = {631--649},
peerreviewed = {Yes},
title = {{On} the integrability of the n-centre problem},
volume = {331},
year = {2005}
}
@article{faucris.109359844,
abstract = {We present numerical and analytical evidence for a first-order phase transition of the ferromagnetic spin chain with partition function Z(fi) = i(fi \Gamma 1)=i(fi) at the inverse temperature fi cr = 2. KEY WORDS: Riemann zeta function, spin chain, phase transition. In a recent paper [6] we established a link between analytic number theory and classical statistical mechanics by interpreting the quotient Z(s) = i(s \Gamma 1)=i(s) of Riemann zeta functions as the partition function of an infinite spin chain with ferromagnetic interactions. For Re(s) ? 2 the quotient has the Dirichlet series representation Z(s) = 1 X n=1 '(n) \Delta n \Gammas (1) where for n 1 the Euler totient function '(n) is defined to be the number of positive integers not exceeding n which are relatively prime to n (that is, '(n) = #fi 2 f1; : : : ;ng j gcd(i;n) = 1g). Now on any half-plane of the form Re(s) ? 2 + ", " ? 0, Z is uniformly approximated by partition functions...},
author = {Knauf, Andreas},
doi = {10.1007/BF01052771},
faupublication = {no},
journal = {Journal of Statistical Physics},
pages = {423--431},
peerreviewed = {Yes},
title = {{Phases} of the {Number}-{Theoretic} {Spin} {Chain}},
volume = {73},
year = {1993}
}
@article{faucris.123231724,
abstract = {We study the dynamics of a charged particle in a planar magnetic field which consists of *n* ≥ 2 disjoint localized peaks. We show that, under mild geometric conditions, this system is semi-conjugated to the full shift on *n* symbols and, hence, carries positive topological entropy.},
author = {Knauf, Andreas and Schulz, Frank and Siburg, Karl Friedrich},
doi = {10.1088/0951-7715/26/3/727},
faupublication = {yes},
journal = {Nonlinearity},
pages = {727--743},
peerreviewed = {Yes},
title = {{Positive} topological entropy for multi-bump magnetic fields},
volume = {26},
year = {2013}
}
@article{faucris.110749364,
abstract = {We derive criteria for the existence of trapped orbits (orbits which are scattering in the past and bounded in the future). Such orbits exist if the boundary of Hill's region is non-empty and not homeomorphic to a sphere. For non-trapping energies we introduce a topological degree which can be non-trivial for low energies, and for Coulombic and other singular potentials. A sum of non-trapping potentials of disjoint support is trapping iff at least two of them have non-trivial degree. For d ≥ 2 dimensions the potential vanishes if for any energy above the non-trapping threshold the classical differential cross section is a continuous function of the asymptotic directions. © Regular and Chaotic Dynamics.},
author = {Knauf, Andreas},
doi = {10.1.1.56.8620},
faupublication = {yes},
journal = {Regular & Chaotic Dynamics},
note = {UnivIS-Import:2015-03-05:Pub.1999.nat.dma.lma6.qualit},
pages = {1-20},
peerreviewed = {Yes},
title = {{Qualitative} {Aspects} of {Classical} {Potential} {Scattering}},
volume = {4},
year = {1999}
}
@article{faucris.118784204,
abstract = {
We review the notion of dynamical entropy by Connes, Narnhofer and Thirring and relate it to Quantum Chaos. A particle in a periodic potential is used as an example. This is worked out in the classical and the quantum mechanical framework, for the single particle as well as for the corresponding gas.

The comparison does not only support the general assertion that quantum mechanics is qualitatively less chaotic than classical mechanics. More specifically, the same dynamical mechanism by which a periodic potential leads to a positive dynamical entropy of the classical particle may *reduce* the dynamical entropy of the quantum gas in comparison to free motion.

},
author = {Benatti, Fabio and Hudetz, Thomas and Knauf, Andreas},
doi = {10.1007/s002200050489},
faupublication = {no},
journal = {Communications in Mathematical Physics},
pages = {607--688},
peerreviewed = {Yes},
title = {{Quantum} chaos and dynamical entropy},
volume = {198},
year = {1998}
}
@article{faucris.110749584,
abstract = {Although quantum tunneling between phase space tori occurs, it is suppressed in the semiclassical limit ℏ ↘ 0 for the Schrödinger equation of a particle in ℝ^{d} under the influence of a smooth periodic potential. In particular this implies that the distribution of quantum group velocities near energy E converges to the distribution of the classical asymptotic velocities near E, up to a term of the order O(1/√E).},
author = {Asch, Joachim and Knauf, Andreas},
doi = {10.1007/s002200050670},
faupublication = {no},
journal = {Communications in Mathematical Physics},
note = {UnivIS-Import:2015-03-05:Pub.1999.nat.dma.lma6.quantu},
pages = {113-128},
peerreviewed = {Yes},
title = {{Quantum} {Transport} on {KAM} {Tori}},
volume = {205},
year = {1999}
}
@article{faucris.210080053,
abstract = {We investigate the existence of resonances for two-center Coulomb systems with arbitrary charges in two dimensions, defining them in terms of generalized complex eigenvalues of a non-selfadjoint deformation of the two-center Schrodinger operator. We construct the resolvent kernels of the operators and prove that they can be extended analytically to the second Riemann sheet. The resonances are then analyzed by means of perturbation theory and numerical methods.},
author = {Seri, Marcello and Knauf, Andreas and Degli Esposti, Mirko and Jecko, Thierry},
doi = {10.1142/S0129055X16500161},
faupublication = {yes},
journal = {Reviews in Mathematical Physics},
keywords = {Resonances;two-center problem;semiclassical analysis},
peerreviewed = {Yes},
title = {{Resonances} in the two-center {Coulomb} systems},
volume = {28},
year = {2016}
}
@article{faucris.117559684,
abstract = {Consider the Schrödinger operator with semiclassical parameter *h*, in the limit where *h* goes to zero. When the involved long-range potential is smooth, it is well known that the boundary values of the operator’s resolvent at a positive energy λ are bounded by *O*(*h* ^{−1}) if and only if the associated Hamilton flow is non-trapping at energy λ. In the present paper, we extend this result to the case where the potential may possess Coulomb singularities. Since the Hamilton flow then is not complete in general, our analysis requires the use of an appropriate regularization.},
author = {Knauf, Andreas and Castella, François and Jecko, Thierry},
doi = {10.1007/s00023-008-0372-x},
faupublication = {yes},
journal = {Annales Henri Poincaré},
pages = {775--815},
peerreviewed = {Yes},
title = {{Semiclassical} resolvent estimates for {Schrödinger} operators with {Coulomb} singularities},
volume = {9},
year = {2008}
}
@article{faucris.123269784,
abstract = {
We introduce and consider the notion of *stable degeneracies* of translation invariant energy functions, taken at spin configurations of a finite Ising model. By this term we mean the lack of injectivity that cannot be lifted by changing the interaction.

We show that besides the symmetry-induced degeneracies, related to spin flip, translation and reflection, there exist additional stable degeneracies, due to more subtle symmetries. One such symmetry is the one of the Singer group of a finite projective plane.

Others are described by combinatorial relations akin to trace identities. Our results resemble traits of the length spectrum for closed geodesics on a Riemannian surface of constant negative curvature. There, stable degeneracy is defined w.r.t. Teichmüller space as parameter space.

},
author = {Knauf, Andreas},
doi = {10.1007/s00220-016-2579-x},
faupublication = {yes},
journal = {Communications in Mathematical Physics},
pages = {1432-0916},
peerreviewed = {Yes},
title = {{Stable} {Degeneracies} for {Ising} {Models}},
year = {2016}
}
@article{faucris.123233484,
abstract = {We analyze a class of stochastically stable quenched measures.We prove that stochastic stability is fully characterized by an infinite family of zero average polynomials in the covariance matrix entries.},
author = {Knauf, Andreas and Contucci, Pierluigi and Bianchi, Alessandra},
doi = {10.1007/s10955-004-5707-5},
faupublication = {yes},
journal = {Journal of Statistical Physics},
keywords = {Disordered systems; stochastic stability; random matrices},
pages = {831--844},
peerreviewed = {Yes},
title = {{Stochastically} stable quenched measures},
volume = {117},
year = {2004}
}
@article{faucris.210080318,
abstract = {For n convex magnetic bumps in the plane, whose boundary has a curvature somewhat smaller than the absolute value of the constant magnetic field inside the bump, we construct a complete symbolic dynamics of a classical particle moving with speed one.},
author = {Knauf, Andreas and Seri, Marcello},
doi = {10.1134/S1560354717040074},
faupublication = {yes},
journal = {Regular & Chaotic Dynamics},
keywords = {magnetic billiards;symbolic dynamics;classical mechanics},
pages = {448-454},
peerreviewed = {Yes},
title = {{Symbolic} {Dynamics} of {Magnetic} {Bumps}},
volume = {22},
year = {2017}
}
@article{faucris.123232384,
abstract = {We show existence and give an implicit formula for the escape rate of the *n*-centre problem of celestial mechanics for high energies. Furthermore we give precise computable estimates of this rate. This exponential decay rate plays an important role especially in semiclassical scattering theory of *n*-atomic molecules. Our result shows that the diameter of a molecule is measurable in a (classical) high-energy scattering experiment.},
author = {Knauf, Andreas and Krapf, Markus},
doi = {10.1007/s11040-010-9073-z},
faupublication = {yes},
journal = {Journal of Mathematical Physics Analysis Geometry},
pages = {159--189},
peerreviewed = {Yes},
title = {{The} escape rate of a molecule},
volume = {13},
year = {2010}
}
@incollection{faucris.110337524,
abstract = {We report on previous and recent work in classical perturbation theory related to the Hannay angles. They are a means of measuring an anholonomy effect in classical mechanics closely corresponding to the Berry phase in quantum mechanics.},
author = {Golin, Simon and Knauf, Andreas and Marmi, Stefano},
booktitle = {Number Theory and Physics},
doi = {10.1007/978-3-642-75405-0{\_}23},
editor = {Jean-Marc Luck, Pierre Moussa, Michel Waldschmidt},
faupublication = {no},
pages = {216-222},
peerreviewed = {unknown},
publisher = {Springer},
series = {Proceedings in Physics},
title = {{The} {Hannay} {Angles} and {Classical} {Perturbation} {Theory}},
volume = {47},
year = {1990}
}
@article{faucris.117567384,
abstract = {The Hannay angles were introduced by Hannay as a means of measuring a holonomy effect in classical mechanics closely corresponding to the Berry phase in quantum mechanics. Using parameter-dependent momentum mappings we show that the Hannay angles are the holonomy of a natural connection. We generalize this effect to non-Abelian group actions and discuss non-integrable Hamiltonian systems. We prove an averaging theorem for phase space functions in the case of general multi-frequency dynamical systems which allows us to establish the almost adiabatic invariance of the Hannay angles. We conclude by giving an application to celestial mechanics.},
author = {Golin, Simon and Knauf, Andreas and Marmi, Stefano},
doi = {10.1007/BF01244019},
faupublication = {no},
journal = {Communications in Mathematical Physics},
pages = {95--122},
peerreviewed = {Yes},
title = {{The} {Hannay} {Angles}: {Geometry}, {Adiabaticity}, and an {Example}},
url = {http://projecteuclid.org/euclid.cmp/1104178683},
volume = {123},
year = {1989}
}
@article{faucris.117564304,
abstract = {We show that a rigorous statistical mechanics description of some Dirichlet series is possible. Using the abstract polymermodel language of statistical mechanics and the polymer expansion theory we characterize the *low* *activity* phase by the suitable exponential decay of the truncated correlation functions},
author = {Contucci, Pierluigi and Knauf, Andreas},
doi = {10.1063/1.531717},
faupublication = {no},
journal = {Journal of Mathematical Physics},
pages = {5458--5475},
peerreviewed = {Yes},
title = {{The} low activity phase of some {Dirichlet} series},
volume = {37},
year = {1996}
}
@article{faucris.109492284,
abstract = {We consider the classical three-dimensional motion in a potential which is the sum of n attracting or repelling Coulombic potentials. Assuming a non-collinear configuration of the n centres, we find a universal behaviour for all energies E above a positive threshold.

Whereas for n=1 there are no bounded orbits, and for n=2 there is just one closed orbit, for n≥3 the bounded orbits form a Cantor set. We analyze the symbolic dynamics and estimate Hausdorff dimension and topological entropy of this hyperbolic set.

Then we set up scattering theory, including symbolic dynamics of the scattering orbits and differential cross section estimates.

The theory includes the n--centre problem of celestial mechanics, and prepares for a geometric understanding of a class of restricted n-{\em body} problems.

To allow for applications in semiclassical molecular scattering, we include an additional smooth (electronic) potential which is arbitrary except its Coulombic decay at infinity. Up to a (optimal) relative error of order 1/E, all estimates are independent of that potential but only depend on the relative positions and strengths of the centres.

Finally we show that different, non-universal, phenomena occur for collinear configurations.},
author = {Knauf, Andreas},
doi = {10.1007/s100970100037},
faupublication = {yes},
journal = {Journal of the European Mathematical Society},
note = {UnivIS-Import:2015-03-09:Pub.2001.nat.dma.lma6.thence},
pages = {1-114},
peerreviewed = {Yes},
title = {{The} n-{Centre} {Problem} of {Celestial} {Mechanics}},
volume = {4},
year = {2001}
}
@article{faucris.122601864,
abstract = {
We consider classical potential scattering. If at energy *E* no orbit is trapped, the Hamiltonian dynamics define an integer-valued topological degree deg(*E*) ≤ 1. This is calculated explicitly for all potentials, and exactly integers ≤1 are shown to occur for suitable potentials.

The non-trapping condition is restrictive in the sense that for a bounded potential it is shown to imply that boundary of Hill's region in configuration space is either empty or homeomorphic to a sphere.

However, in many situations one can decompose a potential into a sum of non-trapping potentials with a non-trivial degree and embed symbolic dynamics of multi-obstacle scattering. This comprises a large number of earlier results, obtained by different authors on multi-obstacle scattering.

},
author = {Knauf, Andreas and Krapf, Markus},
doi = {10.1088/0951-7715/21/9/005},
faupublication = {yes},
journal = {Nonlinearity},
pages = {2023--2041},
peerreviewed = {Yes},
title = {{The} non-trapping degree of scattering},
volume = {21},
year = {2008}
}
@article{faucris.110638704,
abstract = {It is an empirical observation that the Riemann zeta function can be well approximated in its critical strip using the Number-Theoretical Spin Chain. A proof of this would imply the Riemann Hypothesis. Here we relate that question to the one of spectral radii of a family of Markov chains. This in turn leads to the question whether certain graphs are Ramanujan. The general idea is to explain the pseudorandom features of certain number-theoretical functions by considering them as observables of a spin chain of statistical mechanics. In an appendix we relate the free energy of that chain to the Lewis Equation of modular theory.},
author = {Knauf, Andreas},
doi = {10.1007/s002200050441},
faupublication = {no},
journal = {Communications in Mathematical Physics},
note = {UnivIS-Import:2015-03-05:Pub.1998.nat.dma.lma6.thenum},
pages = {703-731},
peerreviewed = {Yes},
title = {{The} {Number}-{Theoretical} {Spin} {Chain} and the {Riemann} {Zeroes}},
volume = {196},
year = {1998}
}
@article{faucris.123236124,
author = {Contucci, Pierluigi and Knauf, Andreas},
doi = {10.1515/form.1997.9.547},
faupublication = {no},
journal = {Forum Mathematicum},
pages = {547--567},
peerreviewed = {Yes},
title = {{The} phase transition of the number-theoretical spin chain},
volume = {9},
year = {1997}
}
@article{faucris.117561664,
abstract = {Among all embedded closed manifolds with positive exterior curvature ≤*k* the ratio between the (*d*-1)-Hausdorff measure of the shadow boundary projection and the volume of *M* ^{ d } is maximized by the sphere of radius 1/*k*.},
author = {Knauf, Andreas},
doi = {10.1007/s002290050045},
faupublication = {no},
journal = {Manuscripta Mathematica},
pages = {519--527},
peerreviewed = {Yes},
title = {{The} size of the shadow boundary projection},
volume = {95},
year = {1998}
}
@article{faucris.117559024,
abstract = {With the help of the representation of SL(2,Z) on the rank two free module over the integer adeles, we define the transition operator of a Markov chain. The real component of its spectrum exhibits a gap, whereas the non-real component forms a circle of radius 1/\sqrt{2}.},
author = {Knauf, Andreas},
doi = {10.1512/iumj.2015.64.5655},
faupublication = {yes},
journal = {Indiana University Mathematics Journal},
pages = {1465--1512},
peerreviewed = {Yes},
title = {{The} spectrum of an adelic {Markov} operator},
volume = {64},
year = {2015}
}
@article{faucris.245474399,
abstract = {The classical Morse theory proceeds by considering sublevel sets f^{−1} (−∞, a] of a Morse function f: M→ℝ, where M is a smooth finite-dimensional manifold. In this paper, we study the topology of the level sets f^{−1}(a) and give conditions under which the topology of f^{−1}(a) changes when passing a critical value. We show that for a general class of functions, which includes all exhaustive Morse functions, the topology of a regular level f^{−1}(a) always changes when passing a single critical point, unless the index of the critical point is half the dimension of the manifold M. When f is a natural Hamiltonian on a cotangent bundle, we obtain more precise results in terms of the topology of the base space. (Counter-)examples and applications to celestial mechanics are also discussed.},
author = {Knauf, Andreas and Martynchuk, Nikolay},
doi = {10.4310/ARKIV.2020.v58.n2.a6},
faupublication = {yes},
journal = {Arkiv For Matematik},
keywords = {Hamiltonian and celestial mechanics; invariant manifolds; Morse theory; Surgery theory; Vector bundles},
note = {CRIS-Team Scopus Importer:2020-11-20},
pages = {333-356},
peerreviewed = {Yes},
title = {{Topology} change of level sets in {Morse} theory},
volume = {58},
year = {2020}
}
@incollection{faucris.117562984,
abstract = {This is a report on recent joint work with J. Asch, and with T. Hudetz and F. Benatti.We consider classical, quantum and semiclassical motion in periodic potentials and prove various results on the distribution of asymptotic velocities.The Kolmogorov-Sinai entropy and its quantum generalization, the Connes-Narnhofer-Thirring entropy, of the single particle and of a gas of noninteracting particles are related.},
author = {Knauf, Andreas},
booktitle = {Séminaire sur les Équations aux Dérivées Partielles, 1996--1997},
faupublication = {no},
pages = {1-11},
peerreviewed = {unknown},
publisher = {École Polytech., Palaiseau},
title = {{Velocity} and entropy of motion in periodic potentials},
url = {http://eudml.org/doc/10921},
volume = {1996-1997},
year = {1997}
}