General informations:

Sunday 8:30 - 12:30, 17:00 - 20:00,
Monday and Tuesday 8:30 - 12:30, 14:00 - 17:45,
Wednesday 8:30 - 13:00.

Map: On this map you find all relevant locations in Regenburg. To the lower right is the railway station (Hauptbahnhof); 150 m towards north on Maximilianstrasse you find the "Altstadtbus", the city bus that takes you close to your hotel "Arch" at Haidplatz 4 (near Bismarckplatz and Arnulfsplatz). The conference site "Haus der Begegnung" is at 200m to the east of the hotel (see more details below).

Below is the schedule of talks and other events. Note that this is a tentative program, just to have something to start from. However, changes may occur and will be announced during the workshop. If you find that you cannot be ready for your talk by the allotted time, or if you are not satisfied with it for other reasons, please let us know. We will try to make everybody happy. But keep in mind that somebody must start, and someone has got to finish!

The conference site is in the historical building "Haus der Begegnung" (Hinter der Grieb 8, same location as Restaurant "Vitus") in the centre of the old town of Regensburg. (Take a flight over the city by clicking on the images!) Besides black board, overhead projector and a beamer, no multimedia will be available there - sorry about that!

All hotel guests will be accommodated in the hotel "Zur Arch" (Haidplatz 4, phone: (+49)(0)941 58660, e-mail: arch@onlinehome.de) at about 3 minutes' walking distance from the conference site. When you register at the hotel "Arch", you will receive these informations, a map of Regensburg, and information how to walk to the conference site. (It's easy enough: exit the hotel, go left around the building into Rote Hahnengasse" and walk straight for about 100 metres. When Rote Hahnengasse turns right, take the little street "Hinter der Grieb" slightly left for another 100 metres; the entrance is under a panel: "Vitus" to your left.)

Warm meals at economic prices will be offered at the conference site (Restaurant "Vitus") during the lunch breaks from 12:30 - 13:45 (Wednesday 13:00 - 14:00). On Sunday, a common little menu (soup and main dish) will be served for 6,40 EUR. On Monday - Wendesday, you may choose between one main dish for 4,90 EUR, a little menu (main dish plus entrée or desert) for 6,40 EUR, or a full menu (all three) for 9,00 EUR. Vegetarian dishes are available. To accelerate the service, you will be asked to order your main dish from a list that will be circulated every morning (eccept Sunday) until 10h.

A professionally guided tour of the old town of Regensburg (in english) will be offered on Sunday, March 21, at 14:00. Duration: about 90 minutes. After this tour, coffee and cake will be offered by the "Vitus" at 16:00.

A common conference dinner will be served on Tuesday, March 23 at 19:00, at the "Leerer Beutel" (Bertoldstr. 9), a restaurant in a historical place at about 20 minutes' walking distance from the conference site. (See the red dot to the right on this map.)
Please notify us if you desire vegetarian food!

• Matthew Cargo (University of California, Berkeley, USA):
  Quantization rules for quantum integral systems via the continuous Maslov method
  We derive quantization conditions for a complete set of n commuting operators in terms of their Weyl symbols. As a simplification, we consider only symbols whose principle symbols are the action variables of a classically integral system. The continuous Maslov method is a variation on Maslov's original approach: we select M points traversing the classical orbit and assign each a convenient coordinate system. At each point, we find local WKB solutions using a variational principle and transport the solutions from one point to the next using a sequence of Heisenberg and metaplectic operators. We keep track of the transported phase at the origin relative to a fixed convention, take M to infinity, and, after closing the loop, obtain a matching condition which, after restoring manifest Sp(2n) invariance, is our quantization rule.
  Talk (PDF, 5.2MB)

• Stephen Creagh (University of Nottingham, UK):
  Uniform multidimensional barrier penetration
  In their simplest form, Maslov indices arise as phases of reflection from one-dimensional turning points and can be calculated by matching locally to a linear potential. In the case of potential barriers, this calculation can be generalised by matching to a parabolic potential and in this guise a uniform treatment can be given of the transition from classical reflection to classical transmission at the top of a barrier. We will discuss multidimensional generalisations of this approach and relate probabilities of transmission to classical geometrical structures and complex dynamics.
  Talk (PDF, 0.63MB)

• Maurice de Gosson (University of Karlskrona, Blekinge, Sweden):
  What is the Maslov index of a symplectic path?
  The interest in the Maslov index has been revived thanks to Gutzwiller's work on quantization of non-integrable systems. Of particular interest in that context is the calculation of the Maslov index of the monodromy matrix of a periodic Hamiltonian orbit. We give a precise combinatorial and topological definition of that index following ideas of Leray, which were further developed by the speaker. We compare our definition and the results it yields with those of other contributors to the topic.
  Talk (PDF, 0.12MB)

• Harald Friedrich (Technical University of Munich, Germany):
  The Maslov index and other potential parameters away from the semiclassical limit
  The range of applicability of the conventional WKB approximation can be substantially extended by allowing the reflection phase at a classical turning point to deviate from the semiclassical expectation pi/2; this corresponds to allowing nonintegral, in general energy dependent Maslov indices away from the semiclassical limit. In many physically important situations, the appropriate Maslov index can be derived from properties of the potential in the Schrödinger equation. With correct values of the Maslov index and other characteristic parameters of the potential, WKB-based methods can be used to derive analytical expressions for physically relevant quantities, which are highly accurate away from the semiclassical limit and often asymptotically exact in the anticlassical or extreme quantum limit of the Schrödinger equation. Physically important applications include near-threshold quantization and quantum reflection by attractive potential tails as occur in the interaction of atoms and molecules with each other and with surfaces. This makes it possible to predict quantum reflection amplitudes which can be observed in experiments with cold atoms.
  Talk (PDF, 5.6MB)

• Michael Gruber (University of Augsburg, Germany):
  On the quantum Hall effect for Dirac fermions
  The QHE for Schrödinger particles (electrons described by a Schrödinger operator) in strong magnetic fields results in the quantisation of conductance. The remarkable stability of this effect can be traced back to its topological nature. We describe a QHE for Dirac fermions which is present even without a magnetic field. In a simple concrete example we analyze the connections with geometry and topology.
  Part of the results are joint work with Marianne Leitner and Ruedi Seiler.

• Martin Gutzwiller (IBM Research Emeritus and Yale University, USA):
  Short introduction to the trace formula
  Talk (PDF, 3.4MB)
  The Compton effect in space and time
  The starting point for all quantum field theories is always formulated in space (x,y,z) and time t, and numerical calculations in quantum chromodynamics are carried out on a lattice of points in (x,y,z,t). A calculation of the Compton effect for scalar fermions and scalar, massless bosons will be presented in space and time. The corresponding Feynman diagram with 4 external and 2 internal vertices is calculated directly in terms of the propagators as functions of the finite, relativistic distances between these vertices. A simple example is worked out for a given time interval T, for a given initial and final spatial distance L between emitters and detectors, and a scattering angle of . The same technique can be used almost identically for the scattering of light or phonons from donor impurities in gallium arsenide in space and time.
  Talk (PDF, 7.5MB)

• Stefan Keppeler (Lund Institute of Technology, Sweden):
  Semiclassical quantisation rules for the Dirac and Pauli equations
  We derive explicit semiclassical quantisation conditions for the Dirac and Pauli equations. We show that the spin degree of freedom yields a contribution, deriving from a non-Abelian Berry phase, which is of the same order of magnitude as the Maslov correction in Einstein-Brillouin- Keller quantisation. In order to obtain this result, a generalisation of the notion of integrability for a certain skew product flow of classical translational dynamics and classical spin precession has to be derived. Among the examples discussed is the relativistic Kepler problem with Thomas precession, whose treatment sheds some light on the amazing success of Sommerfeld's theory of fine structure [A. Sommerfeld, Ann. Phys. (Leipzig) 51, (1916) 1-94, 125-167].
  Talk (PDF, 0.62MB)

• Matthias Lesch (University of Cologne, Germany):
  Maslov index, spectral flow and the eta invariant
  The Maslov index for paths of Lagrangian subspaces is a significant homotopy invariant in symplectic analysis. When dealing with elliptic boundary value problems of geometric differential operators one is naturally led to study the Maslov index also in an infinite-dimensional context. I will explain the Maslov index in this context and show that it is intimately related to the classical winding number and the spectral flow. The latter, as opposed to the other two, does not have a good finite-dimensional analogue. I will discuss various aspects of the spectral flow and finally the significance of the Maslov index for the general splitting formula of the Atiyah-Singer eta invariant..
  Talk (PDF, 0.31MB)

• Robert Littlejohn (University of California, Berkeley, USA):
  Semiclassical torus quantization to higher order in \hbar
  There are several methods known for carrying semiclassical quantization (the Bohr-Sommerfeld or Einstein-Brillouin-Keller rules) to higher order in \hbar. Some apply only to Hamiltonians of the kinetic-plus-potential type, and some are limited to one-dimensional problems. In this talk I will describe an approach that is based on the Wigner-Weyl-Moyal formalism on flat R^{2n} phase spaces, with results that are "metaplectically covariant" (equivariant under linear canonical transformations). We work with integrable systems, defined as those with a complete set of commuting observables. These observables are represented by their Weyl transforms (or "symbols"); the algebra they satisfy is represented by the Moyal *-product. One of these observables (the "Hamiltonian") is assumed to have a symbol with a generic (quadratic) minimum at some point of phase space. The strategy is to perform a unitary transformation on the set of commuting observables that converts their symbols into functions of the n commuting harmonic oscillators (in n degrees of freedom). At lowest order there is a purely classical problem of finding a canonical transformation that transforms the tori of an integrable system in a region of phase space into the tori of the harmonic oscillators. At higher orders one proceeds by a kind of Dyson expansion in a kind of interaction representation. The calculations are not trivial. The order \hbar^2 corrections to the EBK rules will be presented. In the multidimensional case the results are invariant (but not manifestly so) under changes of origin of angle coordinates on the tori. There are unresolved questions of differential geometry in interpreting these results.
  Talk (PDF, 6.1MB)

• Werner Müller (University of Bonn, Germany):
  Inverse scattering, resonances and functional determinants
  In this talk we will discuss some aspects of functional determinants on noncompact manifolds, including Schrödinger operators. The definition of the functional determinant in the noncompact case involves both eigenvalues and resonances. Functional determinants play an important role in questions of inverse spectral and scattering theory on noncompact manifolds. Also questions of the distribution of resonances for chaotic systems and the relation to trace formulae will be discussed.
  Talk (PDF, 0.35MB)

• Paolo Muratore-Ginanneschi (University of Helsinki, Finland):
  On the relation between Maslov indices and functional determinant theory
  Feynman path integrals provide a unified approach to the quantisation of systems with finite and infinite degrees of freedom. Although a rigorous justification of path integral methods is usually hard, often applications have inspired developments in mathematics. The theory of functional determinants offers an example of such developments. Namely, the semiclassical approximation of Feynman path integrals requires the evaluation of the determinant of infinite-dimensional operators with self-adjoint boundary conditions. In my talk I will shorthy review the theory of functional determinants and in particular a general constructive theorem proved by R. Forman [Inv. Math. 88, 447-493 (1987)]. Functional determinants are most conveniently constructed through homotopy transformations. A natural connection then appears between functional determinant and Morse theory. The Morse (Maslov) index of the associated functional operator defines the homotopy class of the functional determinant and in consequence, the phase factor intervening in the semiclassical approximation to Feynman path integrals.
  Talk (PDF, 0.26MB)

• Vesselin Petkov (University of Bordeaux, France):
  Resonances and semiclassical trace formulae
  The semiclassical trace formulae are well known for operators with point spectrum. On the other hand, for Schrödinger type operators with continuous spectrum we expect to define the resonances and to obtain trace formulae connecting the trace of some operators with the resonances. In this direction the spectral shift function (SSF) plays the role of the function counting eigenvalues. In this survey we will present a general approach for the semiclassical trace formulae working for short and long range perturbations, Stark Hamiltonians as well as for Schrödinger operators.
  Talk (PDF, 0.46MB)

• Paolo Piccione (University of Sao Paolo, Brazil, and University of Camerino, Italy):
  "On the Maslov index of Lagrangian paths that are not transversal to the Maslov cycle. Index theorems in the degenerate case"
  The Maslov index of a Lagrangian path, under a certain transversality assumption, is given by an algebraic count of the intersections of the path with a subvariety of the Lagrangian Grassmannian called the Maslov cycle. In this talk we use the notion of "generalized signatures" at a singularity of a smooth curve of symmetric bilinear forms to determine a formula for the computation of the Maslov index in the case of a real analytic path having possibly non transversal intersections. Using this formula we give a general definition of Maslov index for continuous curves in the Lagrangian Grassmannian, both in the finite and in the infinite dimensional (Fredholm) case, and having arbitrary endpoints. Other notions of Maslov index are also considered, like the index for pairs of Lagrangian paths, Kashiwara's triple Maslov index, and Hörmander's four-fold index. We discuss some applications of the theory, with special emphasis on the study of the Jacobi equation along a semi-Riemannian geodesic. In this context, we prove an extension of several versions of the Morse index theorems for geodesics having possibly conjugate endpoints.
  Talk (PDF, 3.2MB)

• Mikhail Pletyukhov (University of Karlsruhe, Germany):
  Canonically invariant calculation of Maslov indices in practical applications
  We discuss the canonically invariant calculation of Maslov indices, appearing in Gutzwiller's trace formula, as proposed by Sugita. We give explicit formulae for its ingredients and test them numerically for periodic orbits in various Hamiltonian systems with mixed dynamics. We discuss the relation of this approach to Gutzwiller's original definition, as further developed by Creagh, Robbins and Littlejohn [Phys. Rev. A 42, 1907 (1990)]. [See also M. Pletyukhov and M. Brack, J. Phys. A 36, 9449 (2003).]
  Talk (PDF, 0.66MB)

• Jonathan Robbins (University of Bristol, UK):
  Maslov indices, singularities of integrable systems, and the periodic Toda chain
  We consider completely integrable Hamiltonian systems in R^{2n} with constants of motion F = (F_1,...,F_n). Singularities are critical points of F. The set of nondegenerate singularities, where rank dF = n-1 and which have definite linear stability, is a codimension-two invariant symplectic submanifold. The Maslov index of a closed curve is shown to be a sum of contributions +/- 2 from the nondegenerate singularities enclosed, the sign depending on the local orientation and stability of the singularities. For one-freedom systems this corresponds to the well-known formula for the Poincaré index of a closed curve as the oriented difference between the number of elliptic and hyperbolic fixed points enclosed. Examples include n-dimensional rotationally symmetric Hamiltonians and simple bifurcations. Our main application is to the n-particle periodic Toda chain. Here we obtain a complete characterisation of the singularities (of all codimensions) and their stabilities in terms of spectral degeneracies of the Lax matrix L. The Maslov index is determined, modulo 2, by products of holonomies of the eigenvectors of L. (Recent work done by J. A. Foxman and J. M. Robbins.)
  Talk (PDF, 0.79MB)

• Ayumu Sugita (University of Kyoto, Japan; presently University of Regensburg, Germany):
  Maslov index of Gutzwiller's trace formula in terms of the phase space path integral
  I will show a canonically invariant derivation of the trace formula using the phase space path integral representation of the partition function. In this derivation, the Maslov index of a periodic orbit appears as the signature of an infinite dimensional quadratic form which represents the second variation of the action around the orbit. Although it is formally defined as (infinity) - (infinity), there is a one-to-one correspondence between the quadratic form and the matrizant (symplectic matrix which represents linearized flow around the periodic orbit), and the Maslov index is determined from the homotopy class of the matrizant. [See also A. Sugita, Ann. Phys. (N.Y.) 288, 277 (2001).]
  Talk (PDF, 6.2MB)

• Gregor Tanner (University of Nottingham, UK):
  Wave chaos in an elastic disc
  We investigate the relation between the elastic wave equation for plane, isotropic bodies and an underlying classical ray dynamics. We study in particular the eigenfrequencies of an elastic disc with free boundaries and their connection to periodic rays inside the circular domain. Even though the problem is separable, wave mixing between the shear and pressure component of the wave field at the boundary leads to an effective stochastic part in the ray dynamics. This introduces phenomena typically associated with classical chaos as for example an exponential increase in the number of periodic orbits. Classically, the problem can be decomposed into an integrable part and a simple binary Markov process. The wave equation can in the high frequency limit be mapped onto a quantum graph. Implications of this result for the level statistics are discussed. We derive furthermore a periodic orbit trace formula starting from the scattering matrix based on the inside-outside duality between eigen-modes and scattering solutions.
  Talk (PDF, 6.8MB)

15.4.2004 (MB)