Online Publications Physics
by Arnold Neumaier
Below are abstracts and downloadable preprints
of my published (and some unpublished) papers in physics and chemistry.
(online versions of my mathematical
complete list of my publications)
For manuscripts with an e-print number, you can also get the latex
source (of some version of the paper)
from an e-print archive such as
For the published version see the references given.
I do not send out paper copies of my manuscripts.
If you have problems downloading, decoding, or printing a file, look at
A theoretical physics FAQ (begun 2004, continually growing)
Contains over 240 sections with explanations
answering questions from theoretical physics, collected from my answers
to postings to various physics discussion groups.
Most topics are related to quantum mechanics, quantum field theory,
renormalization, the measurement problem, randomness, and philosophical
issues in physics.
Classical models for quantum light,
Slides of a lecture given on April 7, 2016 at the University of Linz.
pdf file (350K)
In this lecture, a timeline is traced from Huygens' wave optics to the
modern concept of light according to quantum electrodynamics.
The lecture highlights the closeness of classical concepts and quantum
concepts to a surprising extent. For example, it is shown that the
modern quantum concept of a qubit was already known in 1852 in fully
Classical models for quantum light II,
Slides of a lecture given on April 8, 2016 at the University of Linz.
pdf file (425K)
In this lecture the results of the historical review given in
''Classical models for quantum light''
are utilized to reassess the meaning of observables and stochastic
processes for the classical and quantum description of light.
In particular we discuss the description of partially coherent,
fluctuating light through classical stochastic Maxwell equations
(with uncertainty in the initial conditions only), and look at a
generalization that works for all quantum aspects of arbitrary
Misconceptions about Virtual Particles,
PhysicsForums Insights (April 2016).
This article goes though a number of wikipedia pages and comments on
their misleading statements about virtual particles and Feynman
diagrams. In addition, the article discusses some textbooks on
quantum field theory and the extent to which they contain problematic
formulations about virtual particles.
A. Neumaier and U.K. Deiters,
The characteristic curves of water,
pdf file (365K)
In 1960, E.H. Brown defined a set of characteristic curves (also known
as ideal curves) of pure fluids, along which some thermodynamic
properties match those of an ideal gas. These curves are used for
testing the extrapolation behaviour of equations of state. This work
is revisited, and an elegant representation of the first-order
characteristic curves as level curves of a master function is proposed.
It is shown that Brown's postulate - that these curves are unique and
dome-shaped in a double-logarithmic p,T representation - may fail for
fluids exhibiting a density anomaly. A careful study of the Amagat
curve (Joule inversion curve) generated from the IAPWS-95 reference
equation of state for water reveals the existence of an additional
The Physics of Virtual Particles,
PhysicsForums Insights (March 2016).
In discussions on the internet (including a number of wikipedia pages)
and in books and articles for non-experts in particle physics, there
is considerable confusion about various notions around the concept of
particles of subatomic size, and in particular about the notion of a
virtual particle. This is partly due to misunderstandings in the
terminology used, and partly due to the fact that subatomic particles
manifest themselves only indirectly, thus leaving a lot of leeway for
the imagination to equip these invisible particles with properties,
some of which sound very magical.
The aim of this Insight article is a definition of physical terms
essential for an informed discussion of which of these properties have
a solid basis in physics, and which of these are gross misconceptions
or exaggerations that shouldn't be taken seriously.
U.K. Deiters and A. Neumaier,
Computer simulation of the characteristic curves of pure fluids,
J. Chem. Eng. Data (2016), DOI: 10.1021/acs.jced.6b00133.
pdf file (313K)
Brown's characteristic curves (also known as ideal curves) describe
states at which one thermodynamic property of a real pure fluid matches
that of an ideal gas; these curves can be used for testing the
extrapolation behaviour of equations of state. In this work, some
characteristic curves are computed directly - without recourse to an
equation of state - for some pair potentials by Monte Carlo computer
simulation. The potentials used are an ab-initio potential for argon,
the 1-center Lennard-Jones potential, and a softer pair potential whose
short-range part is in accordance with quantum mechanical predictions.
The influence of the short-distance repulsion on the characteristic
curves is found to be significant even in the 10-100 MPa pressure
Causal Perturbation Theory,
PhysicsForums Insights (June 2015).
Relativistic quantum field theory is notorious for the occurrence of
divergent expressions that must be renormalized by recipes that on
first sight sound very arbitrary and counterintuitive. This Insight
article shows that it doesn't have to be this way!
Analytic representation of critical equations of state,
J. Statist. Phys. 155 (2014), 603-624.
pdf file (460K)
We propose a new form for equations of state (EOS) of
thermodynamic systems in the Ising universality class.
The new EOS guarantees the correct universality and scaling behavior
close to critical points and is formulated in terms of the scaling
fields only -- unlike the traditional Schofield representation, which
uses a parametric form.
Close to a critical point, the new EOS expresses the square of the
strong scaling field as an explicit function of the thermal
scaling field and the dependent scaling field.
A numerical expression is derived, valid close to critical points.
As a consequence of the construction it is shown that the dependent
scaling field can be written as an explicit function of the relevant
scaling fields without causing strongly singular behavior of the
thermodynamic potential in the one-phase region.
Augmented by additional scaling correction fields, the new EOS also
describes the state space further away from critical points.
It is indicated how to use the new EOS to model multiphase fluid
mixtures, in particular for vapor-liquid-liquid equilibrium (VLLE)
where the traditional revised scaling approach fails.
Phenomenological thermodynamics in a nutshell.
This paper gives a concise, mathematically rigorous description of
phenomenological equilibrium thermodynamics for single-phase systems in
the absence of chemical reactions and external forces.
From the formulas provided, it is an easy step to go to various
examples and applications discussed in standard textbooks (such as
those by Callen or Reichl).
A full discussion of global equilibrium would also involve the
equilibrium treatment of multiple phases and chemical reactions.
Since their discussion offers no new aspects compared with
traditional textbook treatments, they are not treated here.
The present phenomenological approach is similar to that of
Callen, who introduces in his well-known thermodynamics book the basic
concepts by means of a few postulates from which everything else
The present setting is a modified version designed to
match the more fundamental approach based on statistical
mechanics. By specifying the kinematical properties of states
outside equilibrium, his informal thermodynamic
stability arguments (which depend on a dynamical assumption close to
equilibrium) can be replaced by rigorous mathematical arguments.
A multi-phase, multi-component critical equation of state,
pdf file (193K)
Realistic equations of state valid in the whole state space of a
multi-component mixture should satisfy at least three important
(i) The Gibbs phase rule holds.
(ii) At low densities, one can deduce a virial equation of state with
the correct multicomponent structure.
(iii) Close to critical points, plait points, and consolute points,
the correct universality and scaling behavior is guaranteed.
This paper discusses semiempirical equations of state for mixtures that
express the pressure as an explicit function of temperature and the
chemical potentials. In the first part, expressions are derived for
the most important thermodynamic quantities. The main result of the
second part is the construction of a large family of equations of
state with the properties (i)--(iii).
A. Neumaier and D. Westra,
Classical and Quantum Mechanics via Lie algebras.
Manuscript (2008, enlarged revision 2011)
pdf file (3165K)
The goal of this book is to present classical mechanics, quantum
mechanics, and statistical mechanics in an almost completely algebraic
setting, thereby introducing mathematicians, physicists, and
engineers to the ideas relating classical and quantum mechanics with
Lie algebras and Lie groups. The book emphasizes the
closeness of classical and quantum mechanics, and the material is
selected in a way to make this closeness as apparent as possible.
Much of the material covered here is not part of standard
textbook treatments of classical or quantum mechanics (or is only
superficially treated there). For physics students who want to
get a broader view of the subject, this book may therefore serve
as a useful complement to standard treatments of quantum mechanics.
Almost without exception, this book is about precise concepts and
exact results in classical mechanics, quantum mechanics, and
statistical mechanics. The structural properties of
mechanics are discussed independent of computational techniques for
obtaining quantitatively correct numbers from the assumptions made.
The standard approximation machinery for calculating from first
principles explicit thermodynamic properties of materials, or
explicit cross sections for high energy experiments can be found in
many textbooks and is not repeated here.
Renormalization without infinities - an elementary tutorial,
pdf file (362K)
Renormalization is an indispensable tool for modern theoretical
physics. At the same time, it is one of the least appealing techniques,
especially in cases where naive formulations result in divergences
that must be cured - a step that is often done in a mathematically
In this paper, it is shown how the renormalization procedure works
both in singular cases where it removes naive divergences and in
regular cases where a naive approach is possible but renormalization
improves the quality of perturbation theory. In fact, one can see
immediately that the singular situation is simply a limiting case of the
After discussing generalities, the paper introduces a large family of
toy examples, defined by special perturbations of an arbitrary
Hamiltonian with a discrete spectrum.
The examples show explicitly many of the renormalization effects
arising in realistic quantum field theories such as quantum
chromodynamics: logarithmic divergences, running couplings,
asymptotic freedom, dimensional transmutation, the renormalization
group, and renormalization scheme dependent results at any order of
Unlike in more realistic theories, everything is derived rigorously
and nonperturbatively in terms of simple explicit formulas. Thus one
can understand in detail how the infinities arise (if they arise) -
namely as an unphysical infinitely sensitive dependence on the bare
coupling constants. One also sees that all spurious infinities are
cured automatically by the same renormalization process that gives
robust physical results in the case where no infinities occur.
Optical models for quantum mechanics,
Slides of a lecture given on February 16, 2010 at the
Institute for Theoretical Physics, University of Giessen.
pdf file (154K)
This lecture (the second of three) discusses work
towards a new, classical view of quantum mechanics.
It is based on an analysis of polarized light,
of the meaning of quantum ensembles in a field theory,
of classical simulations of quantum computing algorithms,
and resulting optical models for the simulation
of quantum mechanics.
In particular, it is shown that
classical second-order stochastic optics
is precisely the quantum mechanics of a single photon,
with all its phenomenological bells and whistles.
Classical and quantum field aspects of light,
Slides of a lecture given on January 29, 2009 at the Institute of
Quantum Optics and Quantum Information of the Austrian Academy of
pdf file (376K)
This lecture (the first of three) discusses foundational problems
on the nature of light revealed by
1. attempts to define a probability concept for photons,
2. quantum models for photons on demands (and their realization
through laser-induced emission by a calcium ion in a cavity),
3. models explaining the photo effect, and
4. Bell-type experiments for single photon nonlocality.
A simple hidden variable experiment,
ps.gz file (170K),
pdf file (96K)
An experiment is described which proves, using single photons only,
that the standard hidden variables assumptions (commonly used to derive
Bell inequalities) are inconsistent with quantum mechanics.
The analysis is very simple and transparent.
In particular, it demonstrates that a classical wave model
for quantum mechanics is not ruled out by experiments demonstrating the
violation of the traditional hidden variable assumptions.
On the foundations of thermodynamics,
ps.gz file (324K),
pdf file (587K)
On the basis of new, concise foundations, this paper establishes
the four laws of thermodynamics, the Maxwell relations, and the
stability requirements for response functions, in a form applicable to
global (homogeneous), local (hydrodynamic) and microlocal (kinetic)
The present, self-contained treatment needs very little formal
machinery and stays very close to the formulas as they are applied
by the practicing physicist, chemist, or engineer.
From a few basic assumptions, the full structure of phenomenological
thermodynamics and of classical and quantum
statistical mechanics is recovered.
Care has been taken to keep the foundations free of subjective aspects
(which traditionally creep in through information or probability).
One might describe the paper as a uniform treatment of the nondynamical
part of classical and quantum statistical mechanics
``without statistics'' (i.e., suitable for the definite descriptions
of single objects) and
``without mechanics'' (i.e., independent of microscopic assumptions).
When enriched by the traditional examples and applications, this paper
may serve as the basis for a course on thermal physics.
Collapse challenge for interpretations of quantum mechanics,
dvi.gz file (7K),
ps.gz file (61K),
pdf file (62K)
living online version (html)
The collapse challenge for interpretations of quantum mechanics
is to build from first principles and your preferred
interpretation a complete, observer-free quantum model of the
described experiment (involving a photon and two screens), together
with a formal analysis that completely explains the experimental result.
The challenge is explained in detail, and discussed in the light
of the Copenhagen interpretation and the decoherence setting.
Mathematik, Physik und Ewigkeit (mit einem Augenzwinkern betrachtet)
Professorenforum-Journal 6 (2005), No. 3, 37--43.
pdf file (116K)
U. Leonhardt and A. Neumaier,
Explicit effective Hamiltonians for general linear quantum-optical
J. Optics B: Quantum Semiclass. Opt. 6 (2004), L1-L4.
dvi.gz file (15K),
ps.gz file (60K),
pdf file (114K),
Linear optical networks are devices that turn classical
incident modes by a linear transformation into outgoing ones.
In general, the quantum version of such transformations may mix
annihilation and creation operators. We derive a simple formula
for the effective Hamiltonian of a general linear quantum network,
if such a Hamiltonian exists.
Otherwise we show how the scattering matrix of the network
is decomposed into a product of three matrices
that can be generated by Hamiltonians.
Quantum field theory as eigenvalue problem,
dvi.gz file (46K),
ps.gz file (139K),
pdf file (281K),
A mathematically well-defined, manifestly covariant theory of
classical and quantum field is given, based on Euclidean Poisson
algebras and a generalization of the Ehrenfest equation, which
implies the stationary action principle.
The theory opens a constructive spectral approach to finding
physical states both in relativistic quantum field theories and for
flexible phenomenological few-particle approximations.
In particular, we obtain a Lorentz-covariant phenomenological
multiparticle quantum dynamics for electromagnetic and gravitational
interaction which provides a representation of the
Poincaré group without negative energy states. The dynamics
reduces in the nonrelativistic limit to the traditional Hamiltonian
multiparticle description with standard Newton and Coulomb forces.
The key that allows us to overcome the traditional problems in
canonical quantization is the fact that we use the algebra of
linear operators on a space of wave functions slightly bigger
than traditional Fock spaces.
P. Frantsuzov, A. Neumaier and V.A. Mandelshtam,
Gaussian resolutions for equilibrium density matrices,
Chem. Phys. Letters 381 (2003), 117-122.
ps.gz file (145K),
pdf file (193K),
A Gaussian resolution method for the computation of
equilibrium density matrices rho(T) for a general
multidimensional quantum problem is presented.
The variational principle applied to the ``imaginary time''
Schroedinger equation provides the
equations of motion for Gaussians in a resolution of rho(T)
described by their width matrix, center and scale factor,
all treated as dynamical variables.
The method is computationally very inexpensive, has
favorable scaling with the system size and is
surprisingly accurate in a wide temperature range,
even for cases involving quantum tunneling.
Incorporation of symmetry constraints,
such as reflection or particle statistics, is also discussed.
Ensembles and experiments in classical and quantum physics,
Int. J. Mod. Phys. B 17 (2003), 2937-2980.
dvi.gz file (71K),
ps.gz file (169K),
pdf file (333K)
A philosophically consistent axiomatic approach to classical and
quantum mechanics is given. The approach realizes a strong
formal implementation of Bohr's correspondence principle.
In all instances, classical and quantum concepts are fully parallel:
the same general theory has a classical realization and a quantum
Extending the `probability via expectation'
approach of Whittle to noncommuting quantities,
this paper defines quantities, ensembles, and experiments as
mathematical concepts and shows how to model complementarity,
uncertainty, probability, nonlocality and dynamics in these terms.
The approach carries no connotations of unlimited repeatability;
hence it can be applied to unique systems such as the universe.
Consistent experiments provide an elegant solution to the reality
problem, confirming the insistence of the orthodox Copenhagen
interpretation on that there is nothing but ensembles,
while avoiding its elusive reality picture.
The weak law of large numbers explains the emergence of classical
properties for macroscopic systems.
Effective Schrödinger equations for nonlocal and/or
dvi.gz file (35K),
ps.gz file (108K),
pdf file (237K),
The projection formalism for calculating effective Hamiltonians and
resonances is generalized to the nonlocal and/or nonhermitian case,
so that it is applicable to the reduction of relativistic systems
and to dissipative systems modeled by an optical potential.
It is also shown
how to recover all solutions of the time-independent
Schrödinger equation in terms of solutions of the effective
Schrödinger equation in the reduced state space and a
Schrödinger equation in a reference state space.
For practical calculations, it is important that the resulting
formulas can be
used without computing any projection operators. This leads to a
modified coupled reaction channel/resonating group method framework
for the calculation of multichannel scattering information.
V.A. Mandelshtam and A. Neumaier,
Further generalization and numerical implementation of pseudo-time
Schrödinger equations for quantum scattering calculations,
J. Theor. Comput. Chemistry 1 (2002), 1-15.
ps.gz file (128K),
pdf file (330K),
We review and further develop the recently introduced numerical
approach for scattering calculations based on a so called
pseudo-time Schrödinger equation, which is in turn a modification
of the damped Chebyshev polynomial expansion scheme.
The method utilizes a special energy-dependent form for the absorbing
potential in the time-independent Schrödinger equation, in which
the complex energy spectrum E_k is mapped to u_k inside the unit disk,
where u_k are the eigenvalues of some explicitly known sparse matrix U.
Most importantly for the numerical implementation, all the
physical eigenvalues u_k are extreme eigenvalues of U, which allows
one to extract these eigenvalues very efficiently by harmonic
inversion of a pseudo-time autocorrelation function
using the filter diagonalization method. The computation of 2T steps
of the autocorrelation function requires only T sparse real
We describe and compare different schemes, effectively corresponding
to different choices of the energy-dependent absorbing potential,
and test them numerically by calculating resonances of the HCO
molecule. Our numerical tests suggest an optimal scheme that provide
accurate estimates for most resonance states.
A. Neumaier and V.A. Mandelshtam,
Pseudo-time Schrödinger equation with absorbing potential for
quantum scattering calculations,
Phys. Rev. Lett. 86 (2001), 5031-5034.
dvi.gz file (62K),
ps.gz file (159K),
pdf file (589K)
The Schrödinger equation with an energy-dependent complex
absorbing potential, associated with a scattering system,
can be reduced for a special choice of the energy-dependence
to a harmonic inversion problem of a discrete pseudo-time
correlation function. An efficient formula for Green's function
matrix elements is also derived.
Since the exact propagation up to time 2t can be done with only t real
matrix-vector products, this gives an unprecedently efficient scheme
for accurate calculations of quantum spectra for possibly very
Bohmian mechanics contradicts quantum mechanics,
dvi.gz file (17K),
ps.gz file (61K),
pdf file (157K)
It is shown that, for a harmonic oscillator in the ground state,
Bohmian mechanics and quantum mechanics predict values of opposite
sign for certain time correlations.
The discrepancy can be explained by the fact that Bohmian mechanics
has no natural way to accomodate the Heisenberg picture, since the
local expectation values that define the beables of the theory depend
on the Heisenberg time being used to define the operators.
Relations to measurement are discussed, too, and are shown to leave no
loophole for claiming that Bohmian mechanics reproduces all
predictions of quantum mechanics exactly.
Quantendesigns - Grundzüge einer nichtkommutativen Designtheorie,
Dissertation, Institut für Mathematik,
Universität Wien, Wien 1999.
(in German; English title:
Quantum designs - foundations of a non-commutative theory of designs)
ps.gz file (202K),
pdf file (557K),
Quantum designs are sets of subspaces, or equivalent sets of
orthogonal projection matrices, in complex finite dimensional
vector spaces with certain properties. These structures are
generalizations of classical t-designs (the special case of pairwise
commuting matrices), spherical designs, complex t-designs and
equi-isoclinic subspaces. All elements of quantum design theory have
a natural interpretation in terms of quantum theory.
Apart from general theory (e.g., absolute and special bounds),
constructions are given for two classes of quantum designs which
generalize the classical balanced incomplete block designs and affine
designs. One of them gives rise to the first known class of infinitely
many complex 2-designs. Also new tight complex 2-designs are
constructed. The constructions have a close analogy to formalisms of
quantum theory in infinite-dimensional vector spaces.
On a realistic interpretation of quantum mechanics,
dvi.gz file (31K),
ps.gz file (84K),
pdf file (192K),
The best mathematical arguments against a realistic
interpretation of quantum mechanics - that gives definite but
partially unknown values to all observables - are analysed and shown
to be based on reasoning that is not compelling.
This opens the door for an interpretation that, while respecting
the indeterministic nature of quantum mechanics, allows to speak of
definite values for all observables at any time that are, however,
only partially measurable.
The analysis also suggests new areas where the foundations of
quantum theory need to be tested.
On the Many-Worlds-Interpretation,
ASCII text file
These comments intend to show that quantum paradoxes are not resolved
by the "many-worlds" interpretation or metatheory of quantum mechanics;
instead, the latter is full of home-made puzzles and ambiguities.
A. Neumaier, W. Huyer and E. Bornberg-Bauer,
Hydrophobicity Analysis of Amino Acids,
Based on a principal component analysis of 47 published attempts to
quantify hydrophobicity in terms of a single scale,
we define a representation of the 20 amino acids as
points in a 3-dimensional hydrophobicity space and display it by means
of a minimal spanning tree. The dominant scale is found to be close to
two scales derived from contact potentials.
A. Neumaier, S. Dallwig, W. Huyer and H. Schichl,
New techniques for the construction of residue potentials for protein
pp. 212-224 in:
Algorithms for Macromolecular Modelling (P. Deuflhard et al., eds.),
Lecture Notes Comput. Sci. Eng. 4, Springer, Berlin 1999.
pdf file (448K),
ps.gz file (159K),
A smooth empirical potential is constructed for use in off-lattice
protein folding studies. Our potential is a function of the amino acid
labels and of the distances between the C(alpha) atoms of a protein.
The potential is a sum of smooth surface potential terms that model
solvent interactions and of pair potentials that are functions of a
distance, with a smooth cutoff at 12 Å.
Techniques include the use of a fully automatic and reliable estimator
for smooth densities, of cluster analysis to group together amino
acid pairs with similar distance distributions, and of quadratic
programming to find appropriate weights with which the various terms
enter the total potential.
For nine small test proteins, the new potential has minima within
1.3-4.7Å of the PDB geometry, with one exception that
has an error of 8.5Å.
Moreover, a nonuniqueness theorem is given that shows that no set of
equilibrium geometries can determine the true effective potential
Molecular modeling of proteins and mathematical prediction of
SIAM Rev. 39 (1997), 407-460.
ps.gz file (220K),
pdf file (490K),
This paper discusses the mathematical formulation of
and solution attempts for the so-called protein folding problem.
The static aspect is concerned with how to
predict the folded (native, tertiary) structure of a protein, given its
sequence of amino acids. The dynamic aspect asks about the possible
pathways to folding and unfolding, including the stability of the
From a mathematical point of view, there are several main sides
to the static problem:
- the selection of an appropriate potential energy function;
- the parameter identification by fitting to experimental data; and
- the global optimization of the potential.
The dynamic problem entails, in addition, the solution of (because of
multiple time scales very stiff) ordinary or
stochastic differential equations (molecular dynamics simulation),
or (in case of constrained
molecular dynamics) of differential-algebraic equations.
A theme connecting the static and dynamic aspect is the determination
and formation of secondary structure motifs.
The present paper gives a self-contained introduction to the
necessary background from physics and chemistry and surveys some of
the literature. It also discusses the various mathematical problems
arising, some deficiencies of the current models and algorithms,
and possible (past and future) attacks to arrive at
solutions to the protein folding problem.
Experiments: Preparation and Measurement,
ps.gz file (49K),
Measurements can be adequately described without reference to
``the collapse of the wave function'' (or to wave functions at all).
The collapse, as far as it occurs (i.e., the convergence of the
density matrix to one that commutes with the Hamiltonian of the
system), is a natural consequence of the reduced description of
macroscopic systems in the thermodynamic limit
since that leads to a dissipative dynamics. However, in the presence
of spin, there is no complete collapse: macroscopic polarization
phenomena remain that need 2-state quantum physics, a fact that
seems to have escaped notice before. Since polarization is
well-understood as a macroscopic phenomenon (no one ever talked about
philosophical problems related to macroscopic polarization!), there
is no reason to consider the microscopic world as essentially different
from the macroscopic world.
From thermodynamics to quantum theory. Part I: Equilibrium.
dvi.gz file (75K),
ps.gz file (165K),
pdf file (457K),
In this paper, an elementary and self-contained axiomatic treatment
is given of equilibrium thermodynamics including fluctuations.
Among other things, this leads to a natural explanation of the Hilbert
space underlying quantum physics, using only a simple quantization
condition related to the third law of thermodynamics.
T. Rage, A. Neumaier and C. Schlier,
Rigorous verification of chaos in a molecular model,
Phys. Rev. E 50 (1994), 2682-2688.
ps.gz file (81K),
The Thiele-Wilson system, a simple model of a linear, triatomic
molecule, has been studied extensively in the past.
The system exhibits complex molecular dynamics including dissociation,
periodic trajectories and bifurcations.
In addition, it has for a long time been suspected to be chaotic,
but this has never been proved with mathematical rigor.
In this paper, we present numerical results that, using interval
methods, rigorously verify the existence of transversal homoclinic
points in a Poincarè map of this system. By a theorem of Smale,
the existence of transversal homoclinic points in a map rigorously
proves its mixing property, i.e., the chaoticity of the system.
A. Neumaier and T. Rage,
Rigorous chaos verification in discrete dynamical systems,
Physica D 67 (1993), 327-346.
Mathematik - Sprache und Werkzeug der Naturwissenschaften
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Computational Mathematics Links
home page (http://www.mat.univie.ac.at/~neum)
Arnold Neumaier (Arnold.Neumaier@univie.ac.at)