# Research

My research is in the fields of dynamical systems theory and semiclassical quantum mechanics (short wavelengths asymptotics). My research is very broad and applied in nature and a general guiding principle is to make an impact on prominent problems in the natural sciences (mainly atomic and molecular physics, celestial mechanics, electron transport problems and optics). In fact I strongly believe that in order to be successful in modern science one cannot afford to have borders between different disciplines. My interest in dynamical systems first of all concerns Hamiltonian systems and has a very geometric flavor. This also relates to semiclassical quantum mechanics which provides techniques to relate quantum states (their wave functions or phase space analogues like Wigner or Husimi functions) to classical phase space structures in the limit where Planck's constant can be considered as small compared to other characteristic scales of the quantum problem (this could, e.g., mean that the quantum mechanical wavelength is short compared to classical lengths scales). This does not only shed light on the quantum problem as such but also provides powerful techniques to compute quantities which are notoriously difficult to compute from ab initio quantum calculations.

The following three topics I have been working on in recent years are representative for my general interests and approaches.

**Transition State Theory.**
The most widely used approach to compute chemical reaction rates is Transition State Theory which was invented by Wigner and others in the 1930's. The main idea is that on the way from reactants to products the system has to pass through a phase space bottleneck which is called the transition state in the chemistry literature. In the context of Hamiltonian systems such bottlenecks are induced by saddle type equilibrium points. The reaction rate is then computed from defining a so called dividing surface in the bottleneck region and compute the rate from the flux through this surface. For this approach to be useful the dividing surface needs to have the property that it is crossed exactly once by trajectories which pass from reactants to products and is not crossed at all by all other trajectories. A violation of this no-recrossing property would lead to an overestimation of the flux and hence of the reaction rate. The constructions of a recrossing-free dividing surface posed a major problem in the development of transition state theory. After chemists like Pechukas, Pollak and others showed that for systems with two degrees of freedom a recrossing-free dividing surface can be constructed from an unstable periodic orbit associated with the saddle in the 1970's it was only recently that it was understood how this can be generalized to systems with three or more degrees of freedom. The role of the periodic orbit is played by a normally hyperbolic invariant manifold (NHIM) which is a codimension-2 sphere in the surface of constant energy. The NHIM forms the equator of a codimension-1 sphere which gives a recrossing-free dividing surface. The NHIM moreover has codimension-1 stable and unstable manifolds. These are of central importance to understand the dynamics of reactions as they channel the trajectories from reactants to products and separate them from non-reactive trajectories. Most importantly for applications these phase space structures can be explicitly constructed from a normal form.
Using the Weyl calculus of pseudo-differential operator we quantized this normal form and developed efficient algorithms to compute quantum reaction rates and the associated Gamov-Siegert resonances. Currently we are extending this approach to the relative equilibria of rotational symmetry reduced N-body systems which is crucial to understand the dynamics of reactions in rotating molecules.

Collaborators: Stephen Wiggins, Roman Schubert and Peter Collins (University of Bristol), Greg Ezra (Cornell University) and Arseni Goussev (Northumbria University).

### Literature

**\cCift\cci, \"U., Waalkens, H. and Broer, H. W.**(2014)*Cotangent bundle reduction and Poincar\'e-Birkhoff normal forms.*. . (URL) (BibTeX)**\cCift\cci, \"U. and Waalkens, H.**(2013)*Reaction Dynamics Through Kinetic Transition States.*. . (URL) (BibTeX)**\cCift\cci, \"U. and Waalkens, H.**(2012)*Phase space structures governing reaction dynamics in rotating molecules.*. . (URL) (BibTeX)**\cCift\cci, \"U. and Waalkens, H.**(2011)*Holonomy-reduced dynamics of triatomic molecules.*. . (URL) (BibTeX)**Goussev, A., Schubert, R., Waalkens, H. and Wiggins, S.**(2010)*The flux-flux correlation function for anharmonic barriers.*. . (URL) (BibTeX)**Schubert, R., Waalkens, H., Goussev, A. and Wiggins, S.**(2010)*Periodic-orbit formula for quantum reactions through transition states.*. . (URL) (BibTeX)**Goussev, A., Schubert, R., Waalkens, H. and Wiggins, S.**(2010)*Quantum Theory of Reactive Scattering in Phase Space.*In Advances in Quantum Chemistry., pages 269-332. (URL) (BibTeX)**Goussev, A., Schubert, R., Waalkens, H. and Wiggins, S.**(2010)*A Periodic Orbit Formula for Quantum Reactions Through Transition States.*In AIP Conf. Proc.., pages 1593-1596. (URL) (BibTeX)**Waalkens, H. and Wiggins, S.**(2010)*Geometric Models of the Phase Space Structures Governing Reaction Dynamics.*. . (URL) (BibTeX)**A. Goussev, R. Schubert, H. Waalkens and S. Wiggins**(2009)*The Quantum Normal Form Approach to Reactive Scattering: The Cumulative Reaction Probability for Collinear Exchange Reactions.*. . (URL) (BibTeX)**Schubert, R., Waalkens, H. and Wiggins, S.**(2009)*A Quantum Version of Wignerâ€™s Transition State Theory.*. . (URL) (BibTeX)**Ezra, G. S., Waalkens, H. and Wiggins, S.**(2009)*Microcanonical rates, gap times, and phase space dividing surfaces.*. . (URL) (BibTeX)**H. Waalkens, R. Schubert and S. Wiggins**(2008)*Wigner's dynamical transition state theory: classical and quantum.*. . (URL) (BibTeX)**R. Schubert, H. Waalkens and S. Wiggins**(2006)*Efficient computation of transition state resonances and reaction rates from a quantum normal form.*. . (URL) (BibTeX)**H. Waalkens, A. Burbanks and S. Wiggins**(2005)*A formula to compute the microcanonical volume of reactive initial conditions in transition state theory.*. . (URL) (BibTeX)**H. Waalkens, A. Burbanks and S. Wiggins**(2005)*Efficient Procedure to Compute the Microcanonical Volume of Initial Conditions that Lead to Escape Trajectories from a Multidimensional Potential Well..*. . (URL) (BibTeX)**H. Waalkens, A. Burbanks and S. Wiggins**(2005)*Escape from planetary neighbourhoods.*. . (URL) (BibTeX)**H. Waalkens and S. Wiggins**(2004)*Direct construction of a dividing surface of minimal flux for multi-degree-of-freedom systems that cannot be recrossed.*. . (URL) (BibTeX)**H. Waalkens, A. Burbanks and S. Wiggins**(2004)*Phase space conduits for reaction in multidimensional systems: HCN isomerization in three dimensions.*. . (URL) (BibTeX)**H. Waalkens, A. Burbanks and S. Wiggins**(2004)*A computational procedure to detect a new type of high-dimensional chaotic saddle and its application to the 3D Hill's problem.*. . (URL) (BibTeX)

**Microlasers.**
Progress in the development of systems and devices utilizing electromagnetic waves for transmitting and
processing information heavily relies on the availability of efficient miniature laser sources. These sources
include semiconductor, crystalline and polymeric microcavity lasers. Such microlasers consist of a microcavity,
with a size in the μm-domain, equipped with active regions, which are pumped either by photo-pumping or by
injecting the carriers from metallic electrodes. These devices have a number of very promising technological
applications like, e.g., `lab on a chip' biosensing.
The simplest microcavity shape is a thin circular microdisk. Circular microresonators (microdisks) are natural
candidates for lasing since some of their modes have extremely high Q-factors (low thresholds). In such modes,
which are called whispering gallery modes, light circulates around the circumference of the disk trapped by total
internal reflection. The serious drawback of microdisk resonators is their isotropic light emission.
In order to obtain a directional output one has to break the rotational symmetry, for
example, by deforming the boundary of the cavity, or placing obstacles (scatterers) inside of the microdisk.
The first approach has been intensively studied by various groups around the world.
Since the cavities are very thin the Maxwell equations of the electromagnetic waves reduce to a scalar wave equation. Formally the problems becomes equivalent to that of an open quantum billiard which makes it possible to use the full machinery of semiclassical quantum mechanics.
We mainly followed the second approach of symmetry breaking and suggested to place a point scatterer inside the microdisk. Mathematically the point scatterer can be described using self-adjoint extension theory which makes the problem to a large extent analytically tractable. We have demonstrated that the presence of the scatterer leads to significant enhancement in the directionality of the outgoing light while preserving the high Q-factors of the whispering gallery modes. We moreover developed a trace formula which allows one to compute the optical modes using the periodic orbits inside of the microdisk and the closed orbits which start and end at the scatterer. There are many future directions. For example, we have been extending the theory to more than one scatterer to realize controlled multidirectional emission.

Collaborators: Martin Sieber and Carl Dettmann (Bristol University) and Gregory Morozov (University of West Scotland).

### Literature

**Morozov, G.V., Sieber, M. and Waalkens, H.**(2015)*Resonances and emission patterns of optical microdisk resonators with scatterers.*In 17th International Conference on Transparent Optical Networks (ICTON), 2015., pages 1-4. (URL) (BibTeX)**Dettmann, C.P., Morozov, G.V., Sieber, M. and Waalkens, H.**(2011)*Microdisk resonators with two point scatterers.*In ICTON: 2011 13th International Conference on Transparent Optical Networks., pages 1-3. (URL) (BibTeX)**Hales, R., Sieber, M. and Waalkens, H.**(2011)*Trace formula for a dielectric microdisk with a point scatterer.*. . (URL) (BibTeX)**Dettmann, C.P., Morozov, G.V., Sieber, M. and Waalkens, H.**(2009)*Unidirectional emission from circular dielectric microresonators with a point scatterer.*. . (URL) (BibTeX)**Dettmann, C.P., Morozov, G.V., Sieber, M. and Waalkens, H.**(2009)*Systematization of All Resonance Modes in Circular Dielectric Cavities.*In ICTON: 2009 11th International Conference on Transparent Optical Networks, Vols 1 and 2., pages 763-766. (URL) (BibTeX)**Dettmann, C.P., Morozov, G.V., Sieber, M. and Waalkens, H.**(2009)*Internal and external resonances of dielectric disks.*. . (URL) (BibTeX)**Dettmann, C.P., Morozov, G.V., Sieber, M. and Waalkens, H.**(2008)*TM and TE Directional Modes of an Optical Microdisk Resonator with a Point Scatterer..*In Proceedings of the 10th International Conference on Transparent Optical Networks (ICTON2008)., pages 65-68. (URL) (BibTeX)**Dettmann, C.P., Morozov, G.V., Sieber, M. and Waalkens, H.**(2008)*Optical Microdisk Resonator with a Small but Finite Size Scatterer..*In Proceedings of the 3rd International Conference on Mathematical Modeling of Wave Phenomena (MMWP08)., pages 287-289. (BibTeX)**C. P. Dettmann, G. V. Morozov, M. Sieber and H. Waalkens**(2008)*Directional emission from an optical microdisk resonator with a point scatterer.*. . (URL) (BibTeX)**Dettmann, C.P., Morozov, G.V., Sieber, M. and Waalkens, H.**(2007)*Far-field emission pattern of a dielectric circular microresonator with a point scatter..*In Ninth International Conference on Transparent Optical Networks (ICTON 2007)., pages 197-200. (URL) (BibTeX)

**Monodromy.**
The only mechanical systems that one can fully solve are those which are integrable. For these systems, one can introduce action-angle variables in terms of which the dynamics is easy: the actions are constant in time and the angles increase with constant rates. Through their adiabatic properties the actions played a prominent role in the early development of quantum mechanics where quantum spectra were obtained from requiring the actions to be integer multiples of Planck's constant. In this context the existence of action-angle variables was already addressed by A. Einstein in his at that time widely ignored paper from 1917. The geometric nature of integrable systems and the existence of action-angle variables was fully understood only much later. It forms the content of the Liouville-Arnold theorem which says that the phase space of a Hamiltonian system which has as many independent constants of motion as degrees of freedom is is foliated by invariant tori on which one can construct action-angle variables. More precisely, this theorem is local in nature: the constants of motion form a so called energy-momentum map. For a regular value of the energy-momentum map, the compact preimage is a torus (or a finite disjoint union of tori) and there exist action-angle variables in a neighbourhood of this torus. Thus locally the phase space has the structure of a trivial torus-bundle over an open neighbourhood of a regular value in the image of the energy-momentum map.
Duistermaat pointed out that globally the torus-bundle over the regular values of the energy-momentum map may be non-trivial. This phenomenon is called monodromy. As a result there may not exist global action-angle variables. Quantisation of a classical system with monodromy leads to quantum monodromy. The fact that the classical actions cannot be globally defined implies that the quantum numbers suffer the same problem.
We have been studying concrete examples which exhibit quantum monodromy like prolate ellipsoidal billiards, Bose-Einstein condensates, the two-center problem which is a model of the hydrogen molecular ion,
general properties like the role of Maslov indices, and generalizations of monodromy to scattering problems where it leads, e.g., to the
non-uniqueness of the quantum phase shift in central scattering.

Collaborators: Holger Dullin (University of Sydney) and Peter H. Richter (Bremen University).

### Literature

**Dullin, Holger R. and Waalkens, Holger**(2018)*Defect in the Joint Spectrum of Hydrogen due to Monodromy.*. . (URL) (BibTeX)**Martynchuk, N. and Waalkens, H.**(2016)*Knauf's Degree and Monodromy in Planar Potential Scattering.*. . (URL) (BibTeX)**H. R. Dullin and H. Waalkens**(2008)*Nonuniqueness of the phase shift in central scattering due to monodromy.*. . (URL) (BibTeX)**H. R. Dullin, J. M. Robbins, H. Waalkens, S. C. Creagh and G. Tanner**(2005)*Maslov indices and monodromy.*. . (URL) (BibTeX)**H. Waalkens, H. R. Dullin and P. H. Richter**(2004)*The problem of two fixed centers: bifurcations, actions, monodromy.*. . (URL) (BibTeX)**Waalkens, H., Junge, A. and Dullin, H. R.**(2003)*Quantum monodromy in the two center-problem.*. . (URL) (BibTeX)**H. Waalkens**(2002)*Quantum monodromy in trapped Bose condensates.*. . (URL) (BibTeX)**H. Waalkens and H. R. Dullin**(2002)*Quantum Monodromy in Prolate Ellipsoidal Billiards.*. . (BibTeX)