@ARTICLE{MDEW2018,

author = {Martynchuk, N. and Dullin, H.R. and Efstathiou, K. and Waalkens, H.}, title = {Scattering invariants in Euler's two-center problem}, journal = {arXiv:1801.09613}, year = {2018}, key = {preprint}, URL = {https://arxiv.org/abs/1801.09613}, abstract = {The problem of two fixed centers was introduced by Euler as early as in 1760. It plays an important role both in celestial mechanics and in the microscopic world. In the present paper we study the spatial problem in the case of arbitrary (both positive and negative) strengths of the centers. Combining techniques from scattering theory and Liouville integrability, we show that this spatial problem has topologically non-trivial scattering dynamics, which we identify as scattering monodromy. The approach that we introduce in this paper applies more generally to scattering systems that are integrable in the Liouville sense.},

}

@ARTICLE{Palasantzas2017,

author = {Tajik, Fatemeh and Sedighi, Mehdi and Masoudi, Amirali and Waalkens, Holger and Palasantzas, George}, title = {Dependence of chaotic actuation dynamics of Casimir oscillators on optical properties and electrostatic effects}, journal = {submitted}, year = {2017}, key = {preprint}, abstract = {With Casimir and electrostatic forces playing a crucial role for the performance and stability of microelectromechanical systems (MEMS), the presence of chaotic behavior, which is often unavoidable, leads to device malfunction due to stiction. Therefore, we investigate here how the optical properties of different materials influence the chaotic behavior of electrostatic torsional MEMS due to changes in magnitude of the Casimir forces and associated torques. We consider the materials Au, which is a good conductor, AIST, which is a phase change material being close to metal in the crystalline state, and finally doped SiC as a very poor conductor. For the conservative systems without friction and driving, there is no chaotic behavior and the analysis of phase portraits and bifurcation diagrams reveal the strong sensitivity of stable actuation dynamics on the material optical properties, while applied electrostatic potentials lead faster to instability and stiction for higher conductivity materials. For the driven systems, the Melnikov method is used to study the presence of chaotic behavior. The results obtained from the Melnikov method are supported by the study of the contours of the transient time to stiction in the phase plane which reveal a substantially increased chaotic behavior for higher conductivity materials, associated with stronger Casimir torques, and applied electrostatic potentials. Because chaotic behavior prohibits long-term prediction of actuation dynamics, its precise understanding is crucial for the design of microdevices.}

}

@ARTICLE{KrajnakWaalkens2018,

author = {Vladimír Krajňák and Holger Waalkens}, title = {The phase space geometry underlying roaming reaction dynamics}, journal = {arXiv:1801.07275 (to appear in J. Math. Chem.)}, year = {2018}, key = {published}, URL ={http://arxiv.org/abs/1801.07275}, abstract = {Recent studies have found an unusual way of dissociation in formaldehyde. It can be characterized by a hydrogen atom that separates from the molecule, but instead of dissociating immediately it roams around the molecule for a considerable amount of time and extracts another hydrogen atom from the molecule prior to dissociation. This phenomenon has been coined roaming and has since been reported in the dissociation of a number of other molecules. In this paper we investigate roaming in Chesnavich's CH$_4^+$ model. During dissociation the free hydrogen must pass through three phase space bottleneck for the classical motion, that can be shown to exist due to unstable periodic orbits. None of these orbits is associated with saddle points of the potential energy surface and hence related to transition states in the usual sense. We explain how the intricate phase space geometry influences the shape and intersections of invariant manifolds that form separatrices, and establish the impact of these phase space structures on residence times and rotation numbers. Ultimately we use this knowledge to attribute the roaming phenomenon to particular heteroclinic intersections.},

}

@ARTICLE{Wiesenfeldetal2016,

author = {Laurent Wiesenfeld and Wing-Fai Thi and Paola Caselli and Alexandre Faure and Luca Bizzocchi and Joao Brandao and Denis Duflot and Eric Herbst and Stephen J. Klippenstein and Tamiki Komatsuzaki and Cristina Puzzarini and Octavio Roncero and Hiroshi Teramoto and Mikito Toda and Ad van der Avoird and Holger Waalkens}, title = {Theory of Gas Phase Scattering and Reactivity for Astrochemistry}, journal = {arXiv:1610.00438}, year = {2016}, url = {http://arxiv.org/abs/1610.00438}, key = {preprint}, abstract = {Because of the very peculiar conditions of chemistry in many astrophysical gases (low densities, mostly low temperatures, kinetics-dominated chemical evolution), great efforts have been devoted to study molecular signatures and chemical evolution. While experiments are being performed in many laboratories, it appears that the efforts directed towards theoretical works are not as strong.

This report deals with the present status of chemical physics/physical chemistry theory, for the qualitative and quantitative understanding of kinetics of molecular scattering, being it reactive or inelastic. By gathering several types of expertise, from applied mathematics to physical chemistry, dialog is made possible, as a step towards new and more adapted theoretical frameworks, capable of meeting the theoretical, methodological and numerical challenges of kinetics-dominated gas phase chemistry in astrophysical environments. A state of the art panorama is presented, alongside present-day strengths and shortcomings. However, coverage is not complete, being limited in this report to actual attendance of the workshop. Some paths towards relevant progress are proposed. } }

@article{DullinWaalkens2018,

title = {Defect in the Joint Spectrum of Hydrogen due to Monodromy}, author = {Dullin, Holger R. and Waalkens, Holger}, journal = {Phys. Rev. Lett. (Editors' Suggestion)}, volume = {120}, issue = {2}, pages = {020507}, numpages = {5}, year = {2018}, month = {Jan}, publisher = {American Physical Society}, key = {published}, doi = {10.1103/PhysRevLett.120.020507}, url = {https://link.aps.org/doi/10.1103/PhysRevLett.120.020507}, abstract = {In addition to the well-known case of spherical coordinates, the Schrödinger equation of the hydrogen atom separates in three further coordinate systems. Separating in a particular coordinate system defines a system of three commuting operators. We show that the joint spectrum of the Hamilton operator, the z component of the angular momentum, and an operator involving the z component of the quantum Laplace-Runge-Lenz vector obtained from separation in prolate spheroidal coordinates has quantum monodromy for energies sufficiently close to the ionization threshold. The precise value of the energy above which monodromy is observed depends on the distance of the focus points of the spheroidal coordinates. The presence of monodromy means that one cannot globally assign quantum numbers to the joint spectrum. Whereas the principal quantum number n and the magnetic quantum number m correspond to the Bohr-Sommerfeld quantization of globally defined classical actions a third quantum number cannot be globally defined because the third action is globally multivalued.}

}

@ARTICLE{BoerWaalkens2017,

AUTHOR = {D. Boer and H. Waalkens}, JOURNAL = {Nederlands Tijdschrift voor Natuurkunde}, TITLE = {Het Groenewoldproduct}, PAGES = {54--57}, VOLUME = {2}, YEAR = {2017}, key = {published},

}

@ARTICLE{BoerWaalkens2017b,

AUTHOR = {D. Boer and H. Waalkens}, JOURNAL = {Dutch Journal of Physics}, TITLE = {The Groenewold product}, VOLUME = {1}, YEAR = {2017}, URL = {https://www.ntvn.nl/magazines/2017-1/The%20Groenewold%20product_1.html}, key = {published},

}

@BOOK{HHvSW2016,

AUTHOR = {H. Han{\ss}mann and I. Hoveijn and S. van Strien and H. Waalkens (editors)}, TITLE = {Dynamics and Geometry}, PUBLISHER = {Indagationes Mathematicae, 27(5): 1029--1336}, KEY = {published}, YEAR = {2016}

}

@article{MartynchukWaalkens2016,

Author = {Martynchuk, N. and Waalkens, H.}, Title = {Knauf's {D}egree and {M}onodromy in {P}lanar

{P}otential {S}cattering },

Journal = {Regular and Chaotic Dynamics}, Volume = {21}, Number = {6}, Pages = {697-706}, Year = 2016, URL = {http://dx.doi.org/10.1134/S1560354716060095}, Key = {published},

}

@ARTICLE{BWSKP2015,

author = {Broer, W. and Waalkens, H. and Svetovoy, V.B. and Knoester, J. and Palasantzas, G.}, title = {Nonlinear actuation dynamics of Casimir oscillators in the presence of surface roughness}, journal = {Phys. Rev. Applied}, year = {2015}, volume = {4}, pages = {054016}, url = {https://journals.aps.org/prapplied/accepted/78070A39Wb81f90780731013c5385d3358dac75db}, key = {published}, abstract = {At separations below 100 nm, Casimir forces strongly influence the actuation dynamics of micro- electromechanical systems (MEMS) in dry vacuum conditions. For a micron size plate oscillating near a surface, which mimics a frequently used setup in experiments with MEMS, we show that the roughness of the surface significantly influences the qualitative dynamics of the oscillator. Via a combination of analytical and numerical methods, it is shown that surface roughness leads to a clear increase of initial conditions associated with chaotic motion, that eventually lead to stiction of the plate on the surface. Since stiction leads to malfunction of MEMS oscillators our results are of central interest for the design of microdevices. Moreover, it is of significance for fundamentally motivated experiments performed with MEMS.}

}

@INPROCEEDINGS{MorozovSieberWaalkens2015,

AUTHOR = {Morozov, G.V. and Sieber, M. and Waalkens, H.}, TITLE = {Resonances and emission patterns of optical microdisk resonators with scatterers}, BOOKTITLE = {17th International Conference on Transparent Optical Networks (ICTON), 2015}, PAGES = {1--4}, YEAR = {2015}, URL = {http://dx.doi.org/10.1109/ICTON.2015.7193642}, key = {published}, abstract = { Optical microdisk resonators with circular boundaries support whispering gallery modes (WGMs) that travel along the boundary. Due to their extremely high Q-factors they have found many applications in, e.g., low-threshold lasing, the sensing of nanoparticles, optical filtering, and nonlinear optics. Recently, it has been demonstrated that the placement of a small scatterer within the microdisk can lead to highly directional modes in various frequency regimes while keeping the high Q-factors. The scatterer lifts the degeneracy of the pairs of clockwise and counter-clockwise propagating modes of the unperturbed circular disk and leads to two standing wave modes. The frequency of the first mode is affected by the scatterer whereas the second mode has a nodal line at the position of the scatterer and its frequency is left unaffected. As a result, more than one scatterer is needed if one wants to change the frequencies of both modes of a degenerate pair. We apply a Green's function method that is based on self-adjoint extension theory for point scatterers to compute the resonant modes with high efficiency and in a systematic way. We show that the placement of two or more scatterers can be used to split the emission into several chosen directions. }

}

@ARTICLE{CiftciWaalkens2014,

author = {\c{C}ift\c{c}i, {\"U}. and Waalkens, H. and Broer, H. W.}, title = {Cotangent bundle reduction and Poincar{\'e}-Birkhoff normal forms}, journal = {Physica D}, volume = {268}, pages = {1--13}, year = {2014}, URL = {http://dx.doi.org/10.1016/j.physd.2013.10.007}, key = {published}, abstract = {In this paper we study a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free Lie group action. The canonical coordinates enable us to compute Poincar{\'e}-Birkhoff normal forms of relative equilibria using standard algorithms. The case of simple mechanical systems with symmetries is studied in detail. As examples we compute Poincar{\'e}-Birkhoff normal forms for a Lagrangian equilateral triangle configuration of a three-body system with a Morse-type potential and the stretched-out configuration of a double spherical pendulum.}

}

@article{CiftciWaalkens2013, Author = {\c{C}ift\c{c}i, {\"U}. and Waalkens, H.}, Journal = {Phys. Rev. Lett.}, Title = {Reaction Dynamics Through Kinetic Transition States}, Year = {2013}, Pages = {233201}, Volume = {110}, KEY = {published}, URL = {http://dx.doi.org/10.1103/PhysRevLett.110.233201}, Abstract = {The transformation of a system from one state to another is often mediated by a bottleneck in the system’s phase space. In chemistry, these bottlenecks are known as transition states through which the system has to pass in order to evolve from reactants to products. The chemical reactions are usually associated with configurational changes where the reactants and products states correspond, e.g., to two different isomers or the undissociated and dissociated state of a molecule or cluster. In this Letter, we report on a new type of bottleneck which mediates kinetic rather than configurational changes. The phase space structures associated with such kinetic transition states and their dynamical implications are discussed for the rotational vibrational motion of a triatomic molecule. An outline of more general related phase space structures with important dynamical implications is given.} }

@ARTICLE{CiftciWaalkens2012,

author = {\c{C}ift\c{c}i, {\"U}. and Waalkens, H.}, title = {Phase space structures governing reaction dynamics in rotating molecules}, journal = {Nonlinearity}, year = {2012}, volume = {25}, pages = {791--892}, URL = {http://dx.doi.org/10.1088/0951-7715/25/3/791}, key = {published}, abstract = {Recently, the phase space structures governing reaction dynamics in Hamiltonian systems have been identified and algorithms for their explicit construction have been developed. These phase space structures are induced by saddle type equilibrium points which are characteristic for reaction type dynamics. Their construction is based on a Poincare ́–Birkhoff normal form. Using tools from the geometric theory of Hamiltonian systems and their reduction, we show in this paper how the construction of these phase space structures can be generalized to the case of the relative equilibria of a rotational symmetry reduced N-body system. As rotations almost always play an important role in the reaction dynamics of molecules, the approach presented in this paper is of great relevance for applications.}

}

@INPROCEEDINGS{DettmannMorozovSieber2011,

AUTHOR = {Dettmann, C.P. and Morozov, G.V. and Sieber, M. and Waalkens, H.}, TITLE = {Microdisk resonators with two point scatterers}, BOOKTITLE = {ICTON: 2011 13th International Conference on Transparent Optical Networks}, PAGES = {1--3}, YEAR = {2011}, URL = {http://dx.doi.org/10.1109/ICTON.2011.5970933}, key = {published}, abstract = { Optical microdisk resonators exhibit modes with extremely high Q-factors. Their low lasing thresholds make circular microresonators good candidates for the realization of miniature laser sources. They have, however, the serious drawback that their light emission is isotropic, which is inconvenient for many applications. In our previous work, we showed that the presence of a point scatterer inside the disk can lead to highly directional modes in various frequency ranges while preserving the high Q-factors. In the present paper we generalize this idea to two point scatterers. The motivation for this work is that the strength of a point scatterer is difficult to control in experiments, and the presence of a second scatterer leads to a higher dimensional parameter space which permits to compensate this deficiency. Similar to the case of a single scatterer in a circular disk, the problem of finding the resonance modes in the presence of two scatterers is to a large extent analytically tractable. }

}

@article{CiftciWaalkens2011,

title={Holonomy-reduced dynamics of triatomic molecules}, author={{\c{C}}ift{\c{c}}i, {\"U}. and Waalkens, H.}, journal={J. Phys. A}, volume={44}, pages={165202}, year={2011}, URL = {http://dx.doi.org/10.1088/1751-8113/44/16/165202}, key = {published}, abstract={Whereas it is easy to reduce the translational symmetry of a molecular system using, e.g., Jacobi coordinates, the situation is much more involved for rotational symmetry. In this paper, we address the latter problem using holonomy reduction. We suggest that the configuration space may be considered as the reduced holonomy bundle with a connection induced by the mechanical connection. Using the fact that for the special case of the three-body problem the holonomy group is SO(2) (as opposed to SO(3) like in systems with more than three bodies), we obtain a holonomy-reduced configuration space of topology R^3_+ x S^1. The dynamics then takes place on the cotangent bundle over the holonomy-reduced configuration space. On this phase space, there is an S1 symmetry action coming from the conserved reduced angular momentum which can be reduced using the standard symplectic reduction method. Using a theorem by Arnold it follows that the resulting symmetry-reduced phase space is again a natural mechanical phase space, i.e. a cotangent bundle. This is different from what is obtained from the usual approach where symplectic reduction is used from the outset. This difference is discussed in some detail, and a connection between the reduced dynamics of a triatomic molecule and the motion of a charged particle in a magnetic field is established. }

}

@ARTICLE{HSW2011,

AUTHOR = {Hales, R. and Sieber, M. and Waalkens, H.}, JOURNAL = {J. Phys. A}, PAGES = {155305}, TITLE = {Trace formula for a dielectric microdisk with a point scatterer}, VOLUME = {44}, YEAR = {2011}, URL = {http://dx.doi.org/10.1088/1751-8113/44/15/155305}, key = {published}, abstract = { Two-dimensional dielectric microcavities are of widespread use in microoptics applications. Recently, a trace formula has been established for dielectric cavities which relates their resonance spectrum to the periodic rays inside the cavity. In this paper, we extend this trace formula to a dielectric disk with a small scatterer. This system has been introduced for microlaser applications, because it has long-lived resonances with strongly directional far field. We show that its resonance spectrum contains signatures not only of periodic rays but also of diffractive rays that occur in Keller’s geometrical theory of diffraction. We compare our results with those for a closed cavity with Dirichlet boundary conditions. }

}

@ARTICLE{GoussevSchubertWaalkensWiggins2010c,

AUTHOR = {Goussev, A. and Schubert, R. and Waalkens, H. and Wiggins, S.}, TITLE = {The flux-flux correlation function for anharmonic barriers}, JOURNAL = {J. Chem. Phys.}, VOLUME = {133}, YEAR = {2010}, PAGES = {244113}, URL = {http://dx.doi.org/10.1063/1.3518425}, key = {published}, abstract = {The flux-flux correlation function formalism is a standard and widely used approach for the computation of reaction rates. In this paper we introduce a method to compute the classical and quantum flux-flux correlation functions for anharmonic barriers essentially analytically through the use of the classical and quantum normal forms. In the quantum case we show that for a general f degree-of-freedom system having an index one saddle the quantum normal form reduces the computation of the flux-flux correlation function to that of an effective one-dimensional anharmonic barrier. The example of the computation of the quantum flux-flux correlation function for a fourth order anharmonic barrier is worked out in detail, and we present an analytical expression for the quantum mechanical microcanonical flux-flux correlation function. We then give a discussion of the short-time and harmonic limits.}

}

@ARTICLE{SchubertWaalkensGoussevWiggins2010,

AUTHOR = {Schubert, R. and Waalkens, H. and Goussev, A. and Wiggins, S.}, TITLE = {Periodic-orbit formula for quantum reactions through transition states}, JOURNAL = {Phys. Rev. A}, VOLUME = {82}, YEAR = {2010}, NUMBER = {1}, PAGES = {012707}, URL = {http://dx.doi.org/10.1103/PhysRevA.82.012707}, key = {published}, abstract = {Transition state theory forms the basis of computing reaction rates in chemical and other systems. Recently, it has been shown how transition state theory can rigorously be realized in phase space by using an explicit algorithm. The quantization has been demonstrated to lead to an efficient procedure to compute cumulative reaction probabilities and the associated Gamov-Siegert resonances. In this paper, these results are used to express the cumulative reaction probability as an absolutely convergent sum over periodic orbits contained in the transition state.}

}

@INPROCEEDINGS{GoussevSchubertWaalkensWiggins2010,

AUTHOR = {Goussev, A. and Schubert, R. and Waalkens, H. and Wiggins, S.}, TITLE = {Quantum Theory of Reactive Scattering in Phase Space}, BOOKTITLE = {Advances in Quantum Chemistry}, VOLUME = {60}, YEAR = {2010}, PAGES = {269--332}, key = {published}, URL = {http://www.elsevier.com/wps/find/bookdescription.cws_home/720862/description#description},

}

@INPROCEEDINGS{GoussevSchubertWaalkensWiggins2010b,

AUTHOR = {Goussev, A. and Schubert, R. and Waalkens, H. and Wiggins, S.}, TITLE = {A Periodic Orbit Formula for Quantum Reactions Through Transition States}, BOOKTITLE = {AIP Conf. Proc.}, VOLUME = {1281}, YEAR = {2010}, PAGES = {1593--1596}, URL = {http://dx.doi.org/10.1063/1.3498119}, key = {published}, abstract = { Transition State Theory forms the basis of computing reaction rates in chemical and other systems. Recently it has been shown how transition state theory can rigorously be realized in phase space using an explicit algorithm. The quantization has been demonstrated to lead to an efficient procedure to compute cumulative reaction probabilities and the associated Gamov-Siegert resonances. These results are used here to derive a formula which expresses the cumulative reaction probability as an absolutely convergent sum over periodic orbits contained in the transition state. }

}

@ARTICLE{WaalkensWiggins2010,

AUTHOR = {Waalkens, H. and Wiggins, S.}, TITLE = {Geometric Models of the Phase Space Structures Governing Reaction Dynamics}, JOURNAL = {Reg. Chaot. Dyn.}, VOLUME = {15}, YEAR = {2010}, PAGES = {1--39}, URL = {http://dx.doi.org/10.1134/S1560354710010016}, key = {published}, abstract = {Hamiltonian dynamical systems possessing equilibria of saddle ? center ? ... ? center stability type display reaction-type dynamics for energies close to the energy of such equilibria; entrance and exit from certain regions of the phase space is only possible via narrow bottlenecks created by the influence of the equilibrium points. In this paper we provide a thorough pedagogical description of the phase space structures that are responsible for controlling transport in these problems. Of central importance is the existence of a Normally Hyperbolic Invariant Manifold (NHIM), whose stable and unstable manifolds have sufficient dimensionality to act as separatrices, partitioning energy surfaces into regions of qualitatively distinct behavior. This NHIM forms the natural (dynamical) equator of a (spherical) dividing surface which locally divides an energy surface into two components (“reactants” and “products”), one on either side of the bottleneck. This dividing surface has all the desired properties sought for in transition state theory where reaction rates are computed from the flux through a dividing surface. In fact, the dividing surface that we construct is crossed exactly once by reactive trajectories, and not crossed by nonreactive trajectories, and related to these properties, minimizes the flux upon variation of the dividing surface.We discuss three presentations of the energy surface and the phase space structures contained in it for 2-degree-of-freedom (DoF) systems in the three-dimensional space ?3, and two schematic models which capture many of the essential features of the dynamics for n-DoF systems. In addition, we elucidate the structure of the NHIM.}

}

@ARTICLE{DettmannMorozovSieber2009b,

AUTHOR = {Dettmann, C.P. and Morozov, G.V. and Sieber, M. and Waalkens, H.}, TITLE = {Unidirectional emission from circular dielectric microresonators with a point scatterer}, JOURNAL = {Phys. Rev. A}, VOLUME = {80}, YEAR = {2009}, NUMBER = {6}, PAGES = {063813}, URL = {http://dx.doi.org/10.1103/PhysRevA.80.063813}, key = {published}, abstract = {Circular microresonators are micron-sized dielectric disks embedded in material of lower refractive index. They possess modes of extremely high Q-factors (low-lasing thresholds), which makes them ideal candidates for the realization of miniature laser sources. They have, however, the disadvantage of isotropic light emission caused by the rotational symmetry of the system. In order to obtain high directivity of the emission while retaining high Q-factors, we consider a microdisk with a pointlike scatterer placed off-center inside of the disk. We calculate the resulting resonant modes and show that some of them possess both of the desired characteristics. The emission is predominantly in the direction opposite to the scatterer. We show that classical ray optics is a useful guide to optimizing the design parameters of this system. We further find that exceptional points in the resonance spectrum influence how complex resonance wave numbers change if system parameters are varied.}

}

@INPROCEEDINGS{DettmannMorozovSieber2009,

AUTHOR = {Dettmann, C.P. and Morozov, G.V. and Sieber, M. and Waalkens, H.}, TITLE = {Systematization of All Resonance Modes in Circular Dielectric Cavities}, BOOKTITLE = {ICTON: 2009 11th International Conference on Transparent Optical Networks, Vols 1 and 2}, PAGES = {763--766}, YEAR = {2009}, URL = {http://dx.doi.org/10.1109/ICTON.2009.5185122}, key = {published}, abstract = {Circular dielectric cavities are key components for the construction of optic microresonators and microlasers. They are one of very few cases where the transcendental equations for complex eigenmodes (resonances) of an open system (dielectric cavity) can be found analytically in an exact manner. The behaviour of those eigenvalues in the small opening limit, i.e. when the refractive index of the cavity goes to infinity, is analysed. The analysis allows one to clearly distinguish between internal (Feshbach) and external (shape) resonance modes for both TM and TE polarizations. As a result, unambiguous azimuthal and radial modal indices are assigned to each internal and external resonance mode.}

}

@ARTICLE{GoussevSchubertWaalkensWiggins2009,

AUTHOR = "A. Goussev and R. Schubert and H. Waalkens and S. Wiggins", TITLE = "The Quantum Normal Form Approach to Reactive Scattering: The Cumulative Reaction Probability for Collinear Exchange Reactions", JOURNAL = "J. Chem. Phys.", VOLUME = "131", PAGES = "144103", YEAR = "2009", URL = {http://dx.doi.org/10.1063/1.3245402}, key = {published}, abstract = {The quantum normal form approach to quantum transition state theory is used to compute the cumulative reaction probability for collinear exchange reactions. It is shown that for heavy-atom systems such as the nitrogen-exchange reaction, the quantum normal form approach gives excellent results and has major computational benefits over full reactive scattering approaches. For light atom systems such as the hydrogen-exchange reaction however, the quantum normal approach is shown to give only poor results. This failure is attributed to the importance of tunneling trajectories in light atom reactions that are not captured by the quantum normal form as indicated by the only very slow convergence of the quantum normal form for such systems.}

}

@ARTICLE{DettmannMorozovSieber2009c,

AUTHOR = {Dettmann, C.P. and Morozov, G.V. and Sieber, M. and Waalkens, H.}, TITLE = {Internal and external resonances of dielectric disks}, JOURNAL = {Europhys. Lett.}, VOLUME = {87}, YEAR = {2009}, NUMBER = {3}, PAGES = {34003}, URL = {http://dx.doi.org/10.1209/0295-5075/87/34003}, key = {published}, abstract = {Circular microresonators (microdisks) are micron size dielectric disks embedded in a material of lower refractive index. They possess modes with complex eigenvalues (resonances) which are solutions of analytically given transcendental equations. The behavior of such eigenvalues in the small opening limit, i.e. when the refractive index of the cavity goes to infinity, is analyzed. This analysis allows one to clearly distinguish between internal (Feshbach) and external (shape) resonant modes for both TM and TE polarizations. This is especially important for TE polarization for which internal and external resonances can be found in the same region of the complex wave number plane. It is also shown that for both polarizations, the internal as well as external resonances can be classified by well-defined azimuthal and radial modal indices.}

}

@ARTICLE{DullinWaalkens2009,

AUTHOR = {Dullin, H.R. and Waalkens, H.}, TITLE = {Dullin and Waalkens reply [to comments by Eiglsperger et al. on the phase shift for 2D central scattering]}, JOURNAL = {Phys. Rev. Lett.}, VOLUME = {102}, YEAR = {2009}, NUMBER = {18}, PAGES = {188902}, URL = {http://dx.doi.org/10.1103/PhysRevLett.102.188902}, key = {published}, abstract = {Dullin and Waalkens reply to comments by Eiglsperger et al. (see ibid., vol. 102, 188901 (2009)). Dullin and Waalkens state that the results of their letter (see ibid., vol. 101, 070405 (2008)) are correct. In the letter they showed that the phase shift for 2D central scattering at a smooth repulsive potential cannot be uniquely defined as a globally smooth function of angular momentum and energy due to a topological obstruction similar to monodromy in bound systems.}

}

@ARTICLE{SchubertWaalkensWiggins2009,

AUTHOR = {Schubert, R. and Waalkens, H. and Wiggins, S.}, TITLE = {A Quantum Version of {W}igner’s Transition State Theory}, JOURNAL = {Few-Body Systems}, VOLUME = {45}, YEAR = {2009}, NUMBER = {2-4}, PAGES = {203--206}, URL = {http://dx.doi.org/10.1007/s00601-009-0037-4}, key = {published}, abstract = {A quantum version of a recent realization of Wigner’s transition state theory in phase space is presented. The theory developed builds on a quantum normal form which locally decouples the quantum dynamics near the transition state to any desired order in (h) over bar. This leads to an explicit algorithm to compute cumulative quantum reaction rates and the associated Gamov-Siegert resonances with high accuracy. This algorithm is very efficient since, as opposed to other approaches, it requires no quantum time propagation.}

}

@ARTICLE { EzraWaalkensWiggins2009, AUTHOR = { Ezra, G. S. and Waalkens, H. and Wiggins, S. }, TITLE = { Microcanonical rates, gap times, and phase space dividing surfaces }, JOURNAL = {J. Chem. Phys.}, VOLUME = {130}, YEAR = {2009}, NUMBER = {16}, PAGES = {164118}, URL = {http://dx.doi.org/10.1063/1.3119365},

key = {published}, abstract = {The general approach to classical unimolecular reaction rates due to Thiele is revisited in light of recent advances in the phase space formulation of transition state theory for multidimensional systems. Key concepts, such as the phase space dividing surface separating reactants from products, the average gap time, and the volume of phase space associated with reactive trajectories, are both rigorously defined and readily computed within the phase space approach. We analyze in detail the gap time distribution and associated reactant lifetime distribution for the isomerization reaction HCN reversible arrow CNH, previously studied using the methods of phase space transition state theory. Both algebraic (power law) and exponential decay regimes have been identified. Statistical estimates of the isomerization rate are compared with the numerically determined decay rate. Correcting the RRKM estimate to account for the measure of the reactant phase space region occupied by trapped trajectories results in a drastic overestimate of the isomerization rate. Compensating but as yet not fully understood trapping mechanisms in the reactant region serve to slow the escape rate sufficiently that the uncorrected RRKM estimate turns out to be reasonably accurate, at least at the particular energy studied. Examination of the decay properties of subensembles of trajectories that exit the HCN well through either of two available symmetry related product channels shows that the complete trajectory ensemble effectively attains the full symmetry of the system phase space on a short time scale t less than or similar to 0.5 ps, after which the product branching ratio is 1:1, the "statistical" value. At intermediate times, this statistical product ratio is accompanied by nonexponential (nonstatistical) decay. We point out close parallels between the dynamical behavior inferred from the gap time distribution for HCN and nonstatistical behavior recently identified in reactions of some organic molecules.}

}

@ARTICLE{Waalkens2009,

AUTHOR = {Waalkens, H.}, TITLE = {Quantum transition state theory}, JOURNAL = {Bulletin Am. Phys. Soc.}, VOLUME = {51}, YEAR = {2009}, key = {published}, abstract = {The main idea of Wigner’s transition state theory (TST) is to compute reaction rates from the flux through a dividing surface placed between reactants and products. In order not to overestimate the rate the dividing surface needs to have the no- recrossing property, i.e. reactive trajectories cross the dividing surface exactly once, and nonreactive trajectories do not cross it at all. The long standing problem of how to construct such a diving surface for multi-degree-of-freedom systems was solved only recently using ideas from dynamical systems theory. Here a normal form allows for a local decoupling of the classical dynamics which leads to the explicit construction of the phase space structures that govern the reaction dynamics through transition states. The dividing surface is spanned by a normally hyperbolic manifold which is the mathematical manifestation of the transition state as an unstable invariant subsystem of one degree of freedom less than the full system. The mere existence of a quantum version of TST is discussed controversially in the literature. The key isssue is the presence of quantum mechanical tunneling which prohibits the existence of a local theory analogous to the classical case. Various approaches have been devloped to overcome this problem by propagating quantum wavefunctions through the transition state region. These approaches have in common that they are computationally very expensive which seriously limits their applicability. In contrast the approach by Roman Schubert, Stephen Wiggins and myself is local in nature. A quantum normal form allows us to locally decouple the quantum dynamics to any desired order in Planck’s constant. This yields not only the location of the scattering and resonance wavefunctions relative to the classical phase space structures, but also leads to very efficient algorithms to compute cumulative reaction probabilities and Gamov-Siegert resonances which are the quantum imprints of the transition state.}

}

@ARTICLE{HalesWaalkens2009,

AUTHOR = {Hales, R. and Waalkens, H.}, TITLE = {Classical and quantum transport through entropic barriers modeled by hardwall hyperboloidal constrictions}, JOURNAL = {Ann. Phys. (NY)}, VOLUME = {324}, YEAR = {2009}, NUMBER = {7}, PAGES = {1408--1451}, URL = {http://dx.doi.org/10.1016/j.aop.2009.01.010}, key = {published}, abstract = {We study the quantum transport through entropic barriers induced by hardwall constrictions of hyperboloidal shape in two and three spatial dimensions. Using the separability of the Schrodinger equation and the classical equations of motion for these geometries, we study in detail the quantum transmission probabilities and the associated quantum resonances, and relate them to the classical phase structures which govern the transport through the constrictions. These classical phase structures are compared to the analogous structures which, as has been shown only recently, govern reaction type dynamics in smooth systems. Although the systems studied in this paper are special due their separability they can be taken as a guide to study entropic barriers resulting from constriction geometries that lead to non-separable dynamics.}

}

@INPROCEEDINGS{DettmannMorozovSieber2008,

AUTHOR = {Dettmann, C.P. and Morozov, G.V. and Sieber, M. and Waalkens, H.}, TITLE = {TM and TE Directional Modes of an Optical Microdisk Resonator with a Point Scatterer.}, BOOKTITLE = {Proceedings of the 10th International Conference on Transparent Optical Networks (ICTON2008)}, PAGES = {65--68}, YEAR = {2008}, volume = {4}, key = {published}, URL = {http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=4598735&isnumber=4598708},

}

@INPROCEEDINGS{DettmannMorozovSieber2008b,

AUTHOR = {Dettmann, C.P. and Morozov, G.V. and Sieber, M. and Waalkens, H.}, TITLE = {Optical Microdisk Resonator with a Small but Finite Size Scatterer.}, BOOKTITLE = {Proceedings of the 3rd International Conference on Mathematical Modeling of Wave Phenomena (MMWP08)}, PAGES = {287--289}, YEAR = {2008}, key = {published}, abstract = {Circular microresonators (microdisks) are natural can- didates for the realization of low-threshold miniature laser sources since some of their modes have extremely high Q-factors (low thresholds). In those modes, which are called whispering gallery modes, light cir- culates around the circumference of the disk trapped by total internal reflection. Although the microdisk cavities can provide ultra-low threshold lasing, their applicability faces the problem of isotropic light emis- sion which is due to the rotational symmetry of the system. Contrary to usual procedure, where a geo- metric deformation of the microdisk boundary is used to break the symmetry and, as a result, to achieve the output directionality, we propose a scenario inducing rotational symmetry breaking by placing a small but finite size circular scatterer inside the microdisk itself. We calculate positions of the new resonant modes and show that some of them possess clear emission direc- tionality while preserving high Q-factors.}

}

@ARTICLE{DullinWaalkens08,

AUTHOR = {H. R. Dullin and H. Waalkens}, TITLE = {Nonuniqueness of the phase shift in central scattering due to monodromy}, JOURNAL = {Phys. Rev. Lett.}, VOLUME = 101, PAGES = {070405}, YEAR = 2008, URL = {http://dx.doi.org/10.1103/PhysRevLett.101.070405}, key = {published}, abstract = {Scattering at a central potential is completely characterized by the phase shifts which are the differences in phase between outgoing scattered and unscattered partial waves. In this Letter, it is shown that, for 2D scattering at a repulsive central potential, the phase shift cannot be uniquely defined due to a topological obstruction which is similar to monodromy in bound systems.}

}

@ARTICLE{DettmannMorozovSieberWaalkens08,

AUTHOR = {C. P. Dettmann and G. V. Morozov and M. Sieber and H. Waalkens}, TITLE = {Directional emission from an optical microdisk resonator with a point scatterer}, JOURNAL = {Europhys. Lett.}, VOLUME = {82}, PAGES = {34002}, YEAR = 2008, URL = {http://dx.doi.org/10.1209/0295-5075/82/34002}, key = {published}, abstract = {We present a new design of dielectric microcavities supporting modes with large quality factors and highly directional light emission. The key idea is to place a point scatterer inside a dielectric circular microdisk. We show that, depending on the position and strength of the scatterer, this leads to strongly directional modes in various frequency regions while preserving the high Q-factors reminiscent of the whispering gallery modes of the microdisk without scatterer. The design is very appealing due to its simplicity, promising a cleaner experimental realisation than previously studied microcavity designs on the one hand and analytic tractability based on Green's function techniques and self-adjoint extension theory on the other.

} }

@ARTICLE{WaalkensSchubertWiggins08,

AUTHOR = {H. Waalkens and R. Schubert and S. Wiggins}, TITLE = {{W}igner's dynamical transition state theory: classical and quantum}, JOURNAL = {Nonlinearity}, VOLUME = {21}, PAGES = {R1-R118}, YEAR = 2008, URL = {http://dx.doi.org/10.1088/0951-7715/21/1/R01}, key = {published}, abstract = {We develop Wigner's approach to a dynamical transition state theory in phase space in both the classical and quantum mechanical settings. The key to our development is the construction of a normal form for describing the dynamics in the neighbourhood of a specific type of saddle point that governs the evolution from reactants to products in high dimensional systems. In the classical case this is the standard Poincaré–Birkhoff normal form. In the quantum case we develop a normal form based on the Weyl calculus and an explicit algorithm for computing this quantum normal form. The classical normal form allows us to discover and compute the phase space structures that govern classical reaction dynamics. From this knowledge we are able to provide a direct construction of an energy dependent dividing surface in phase space having the properties that trajectories do not locally 're-cross' the surface and the directional flux across the surface is minimal. Using this, we are able to give a formula for the directional flux through the dividing surface that goes beyond the harmonic approximation. We relate this construction to the flux–flux autocorrelation function which is a standard ingredient in the expression for the reaction rate in the chemistry community. We also give a classical mechanical interpretation of the activated complex as a normally hyperbolic invariant manifold (NHIM), and further describe the structure of the NHIM. The quantum normal form provides us with an efficient algorithm to compute quantum reaction rates and we relate this algorithm to the quantum version of the flux–flux autocorrelation function formalism. The significance of the classical phase space structures for the quantum mechanics of reactions is elucidated by studying the phase space distribution of scattering states. The quantum normal form also provides an efficient way of computing Gamov–Siegert resonances. We relate these resonances to the lifetimes of the quantum activated complex. We consider several one, two and three degree-of-freedom systems and show explicitly how calculations of the above quantities can be carried out. Our theoretical framework is valid for Hamiltonian systems with an arbitrary number of degrees of freedom and we demonstrate that in several situations it gives rise to algorithms that are computationally more efficient than existing methods.}

}

@INPROCEEDINGS{DettmannMorozovSieber2007,

AUTHOR = {Dettmann, C.P. and Morozov, G.V. and Sieber, M. and Waalkens, H.}, TITLE = {Far-field emission pattern of a dielectric circular microresonator with a point scatter.}, BOOKTITLE = {Ninth International Conference on Transparent Optical Networks (ICTON 2007)}, PAGES = {197--200}, YEAR = {2007}, key = {published}, URL = {http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=04296377},

}

@ARTICLE{SchubertWaalkensWiggins06,

AUTHOR = {R. Schubert and H. Waalkens and S. Wiggins}, TITLE = {Efficient computation of transition state resonances and reaction rates from a quantum normal form}, VOLUME = {96}, JOURNAL = {Phys. Rev. Lett.}, PAGES = {218302}, YEAR = 2006, URL = {http://dx.doi.org/10.1103/PhysRevLett.96.218302}, key = {published}, abstract = {A quantum version of a recent formulation of transition state theory in phase space is presented. The theory developed provides an algorithm to compute quantum reaction rates and the associated Gamov- Siegert resonances with very high accuracy. The algorithm is especially efficient for multi-degree-of- freedom systems where other approaches are no longer feasible.}

}

@ARTICLE{WaalkensBurbanksWiggins05c,

AUTHOR = {H. Waalkens and A. Burbanks and S. Wiggins}, TITLE = {A formula to compute the microcanonical volume of reactive initial conditions in transition state theory}, JOURNAL = {J. Phys. A}, VOLUME = 38, PAGES = {L759-L768}, YEAR = 2005, URL = {http://dx.doi.org/10.1088/0305-4470/38/45/L03}, key = {published}, abstract = {We present the formal proof of a procedure to compute the phase-space volume of initial conditions for trajectories that, for a constant energy, escape or 'react' from a multi-dimensional potential well with one or several exit/entrance channels. The procedure relies on a phase-space formulation of transition state theory. It gives the volume of reactive initial conditions as the sum over the exit/entrance channels where each channel contributes by the product of the phase-space flux associated with the channel and the mean residence time in the well of those trajectories which escape through the channel. An example is given to demonstrate the computational efficiency of the procedure.}

}

@ARTICLE{WaalkensBurbanksWiggins06,

AUTHOR = {H. Waalkens and A. Burbanks and S. Wiggins}, TITLE = {Efficient Procedure to Compute the Microcanonical Volume of Initial Conditions that Lead to Escape Trajectories from a Multidimensional Potential Well.}, VOLUME = {95}, JOURNAL = {Phys. Rev. Lett.}, PAGES = {084301}, YEAR = 2005, URL = {http://dx.doi.org/10.1103/PhysRevLett.95.084301}, key = {published}, abstract = {A procedure is presented for computing the phase space volume of initial conditions for trajectories that escape or ‘‘react’’ from a multidimensional potential well. The procedure combines a phase space transition state theory, which allows one to construct dividing surfaces that are free of local recrossing and that minimize the directional flux, and a classical spectral theorem. The procedure gives the volume of reactive initial conditions in terms of a sum over each entrance channel of the well of the product of the phase space flux across the dividing surface associated with the channel and the mean residence time in the well of trajectories which enter through the channel. This approach is illustrated for HCN isomerization in three dimensions, for which the method is several orders of magnitude more efficient than standard Monte Carlo sampling.}

}

@ARTICLE{WaalkensBurbanksWiggins05b,

AUTHOR = {H. Waalkens and A. Burbanks and S. Wiggins}, TITLE = {Escape from planetary neighbourhoods}, JOURNAL = {Mon. Not. R. Astron. Soc.}, VOLUME = 361, PAGES = {763-775}, YEAR = 2005, URL = {http://dx.doi.org/10.1111/j.1365-2966.2005.09237.x}, key = {published}, abstract = {In this paper we use recently developed phase-space transport theory coupled with a so-called classical spectral theorem to develop a dynamically exact and computationally efficient procedure for studying escape from a planetary neighbourhood. The ‘planetary neighbourhood’ is a bounded region of phase space where entrance and escape are only possible by entering or exiting narrow ‘bottlenecks’ created by the influence of a saddle point. The method therefore immediately applies to, for example, the circular restricted three-body problem and Hill's lunar problem (which we use to illustrate the results), but it also applies to more complex, and higher-dimensional, systems possessing the relevant phase-space structure. It is shown how one can efficiently compute the mean passage time through the planetary neighbourhood, the phase-space flux in, and out, of the planetary neighbourhood, the phase-space volume of initial conditions corresponding to trajectories that escape from the planetary neighbourhood, and the fraction of initial conditions in the planetary neighbourhood corresponding to bound trajectories. These quantities are computed for Hill's problem. We study the dependence of the proportions of these quantities on energy and dimensionality (two-dimensional planar and three-dimensional spatial Hill's problem). The methods and quantities presented are of central interest for many celestial and stellar dynamical applications such as, for example, the capture and escape of moons near giant planets, the formation of binaries in the Kuiper belt and the escape of stars from star clusters orbiting about a galaxy.}

}

@ARTICLE{Waalkens05,

AUTHOR = {H. Waalkens}, TITLE = {Quantized conductance through an asymmetric narrow constriction in a three-dimensional electron gas}, JOURNAL = {Phys. Rev. B}, VOLUME = 71, PAGES = {035335}, YEAR = 2005, URL = {http://dx.doi.org/10.1103/PhysRevB.71.035335}, key = {published}, abstract = {The quantized ballistic transmission of a three-dimensional electron gas through a narrow constriction modeled by an asymmetric hyperboloid is studied. The conductance as a function of voltage jumps by integer multiples of e2/???????? each time a new transition channel opens in the plane of narrowest restriction, which has the shape of an ellipse. There are two different modes ??“whispering gallery modes” and “bouncing ball modes”?? which lead to different conductance steps. The smoothing of the conductance steps is discussed in terms of phase space tunneling through a dynamical barrier. A comprehensive interpretation of the quantum mechanical results is obtained from relating them to the corresponding classical motions.}

}

@ARTICLE{DullinRobbinsWaalkensCreaghTanner05,

AUTHOR = {H. R. Dullin and J. M. Robbins and H. Waalkens and S. C. Creagh and G. Tanner}, TITLE = {{M}aslov indices and monodromy}, JOURNAL = {J. Phys. A}, VOLUME = 38, PAGES = {L443-L447}, YEAR = 2005, URL = {http://dx.doi.org/10.1088/0305-4470/38/24/L02}, key = {published}, abstract = {We prove that for a Hamiltonian system on a cotangent bundle that is Liouville-integrable and has monodromy the vector of Maslov indices is an eigenvector of the monodromy matrix with eigenvalue 1. As a corollary, the resulting restrictions on the monodromy matrix are derived.}

}

@ARTICLE{WaalkensWiggins04,

AUTHOR = {H. Waalkens and S. Wiggins}, TITLE = {Direct construction of a dividing surface of minimal flux for multi-degree-of-freedom systems that cannot be recrossed}, JOURNAL = {J. Phys. A}, VOLUME = 37, PAGES = {L435-L445}, YEAR = 2004, URL = {http://dx.doi.org/10.1088/0305-4470/37/35/L02}, key = {published}, abstract = {The fundamental assumption of transition state theory is the existence of a dividing surface having the property that trajectories originating in reactants (resp. products) must cross the surface only once and then proceed to products (resp. reactants). Recently it has been shown (Wiggins et al (2001) Phys. Rev. Lett. 86 5478; Uzer et al (2002) Nonlinearity 15 957) how to construct a dividing surface in phase space for Hamiltonian systems with an arbitrary (finite) number of degrees of freedom having the property that trajectories only cross once locally. In this letter we provide an argument showing that the flux across this dividing surface is a minimum with respect to certain types of variations of the dividing surface.}

}

@ARTICLE{WaalkensBurbanksWigginsb04,

AUTHOR = {H. Waalkens and A. Burbanks and S. Wiggins}, TITLE = {Phase space conduits for reaction in multidimensional systems: {HCN} isomerization in three dimensions}, JOURNAL = {J. Chem. Phys.}, VOLUME = 121, NUMBER = {13}, PAGES = {6207-6225}, YEAR = 2004, URL = {http://dx.doi.org/10.1063/1.1789891}, key = {published}, abstract = {The three-dimensional hydrogen cyanide/isocyanide isomerization problem is taken as an example to present a general theory for computing the phase space structures which govern classical reaction dynamics in systems with an arbitrary ??finite?? number of degrees of freedom. The theory, which is algorithmic in nature, comprises the construction of a dividing surface of minimal flux which is locally a ‘‘surface of no return.’’ The theory also allows for the computation of the global phase space transition pathways that trajectories must follow in order to react. The latter are enclosed by the stable and unstable manifolds of a so-called normally hyperbolic invariant manifold (NHIM). A detailed description of the geometrical structures and the resulting constraints on reaction dynamics is given, with particular emphasis on the three degrees of freedom case. A procedure is given which uses these structures to compute orbits homoclinic to, and heteroclinic between, NHIMs. The role of homoclinic and heteroclinic orbits in global recrossings of dividing surfaces and transport in complex systems is explained. The complete description provided here is inherently one within phase space; it cannot be inferred from a configuration space picture. A complexification of the classical phase space structures to incorporate quantum effects is also discussed. The results presented here call into question certain assumptions routinely made on the global dynamics; this paper provides methods that enable one to understand and quantify the phase space dynamics of reactions without making such assumptions.},

}

@ARTICLE{WaalkensBurbanksWiggins04,

AUTHOR = {H. Waalkens and A. Burbanks and S. Wiggins}, TITLE = {A computational procedure to detect a new type of high-dimensional chaotic saddle and its application to the 3{D} {H}ill's problem}, JOURNAL = {J. Phys. A}, VOLUME = 37, PAGES = {L257-L265}, YEAR = 2004, URL = {http://dx.doi.org/10.1088/0305-4470/37/24/L04}, key = {published}, abstract = {A computational procedure that allows the detection of a new type of high- dimensional chaotic saddle in Hamiltonian systems with three degrees of freedom is presented. The chaotic saddle is associated with a so-called normally hyperbolic invariant manifold (NHIM). The procedure allows us to compute appropriate homoclinic orbits to the NHIM from which we can infer the existence of a chaotic saddle. It also allows us to detect heteroclinic connections between different NHIMs. NHIMs control the phase space transport across an equilibrium point of saddle-centre- · · · -centre stability type, which is a fundamental mechanism for chemical reactions, capture and escape, scattering, and, more generally, ‘transformation’ in many different areas of physics. Consequently, the presented methods and results are of broad interest. The procedure is illustrated for the spatial Hill’s problem which is a well-known model in celestial mechanics and which gained much interest, e.g. in the study of the formation of binaries in the Kuiper belt.}

}

@ARTICLE{WaalkensDullinRichter04,

AUTHOR = {H. Waalkens and H. R. Dullin and P. H. Richter}, TITLE = {The problem of two fixed centers: bifurcations, actions, monodromy}, JOURNAL = {Physica D}, VOLUME = 196, NUMBER = {3-4}, PAGES = {265-310}, YEAR = 2004, URL = {http://dx.doi.org/10.1016/j.physd.2004.05.006}, key = {published}, abstract = {A comprehensive analysis of the Euler–Jacobi problem of motion in the field of two fixed attracting centers is given, first classically and then quantum mechanically in semiclassical approximation. The system was originally studied in the context of celestial mechanics but, starting with Pauli’s dissertation, became a model for one-electron molecules such as H 2+ (symmetric case of equal centers) or HHe2+ (asymmetric case of different centers). The present paper deals with arbitrary relative strength of the two centers and considers separately the planar and the three-dimensional problems. All versions represent non-trivial examples of integrable dynamics and are studied here from the unifying point of view of the energy–momentum mapping from phase space to the space of integration constants. The interesting objects are the critical values of this mapping, i.e., its bifurcation diagram, and their pre-images which organize the foliation of phase space into Liouville–Arnold tori. The classical analysis culminates in the explicit derivation of the action variable representation of iso-energetic surfaces. The attempt to identify a system of global actions, smoothly dependent on the integration constants wherever these are non-critical, leads to the detection of monodromy of a special kind which is here described for the first time. The classical monodromy has its counterpart in the quantum version of the two-center problem where it prevents the assignments of unique quantum numbers even though the system is separable.}

}

@ARTICLE{WaalkensJungeDullin04,

AUTHOR = {Waalkens, H. and Junge, A. and Dullin, H. R.}, TITLE = {Quantum monodromy in the two center-problem}, JOURNAL = {J. Phys. A}, VOLUME = 36, PAGES = {L307-L314}, YEAR = 2003, URL = {http://dx.doi.org/10.1088/0305-4470/36/20/103}, key = {published}, abstract = {Using modern tools from the geometric theory of Hamiltonian systems it is shown that electronic excitations in diatoms which can be modelled by the two-centre problem exhibit a complicated case of classical and quantum monodromy. This means that there is an obstruction to the existence of global quantum numbers in these classically integrable systems. The symmetric case of H+2 and the asymmetric case of H He++ are explicitly worked out. The asymmetric case has a non-local singularity causing monodromy. It coexists with a second singularity which is also present in the symmetric case. An interpretation of monodromy is given in terms of the caustics of invariant tori. }

}

@ARTICLE{Waalkens2001b,

AUTHOR = {H. Waalkens}, TITLE = {Quantum monodromy in trapped {B}ose condensates}, JOURNAL = {Europhys. Lett.}, VOLUME = {58}, PAGES = {162-168}, YEAR = 2002, URL = {http://dx.doi.org/10.1209/epl/i2002-00619-7}, key = {published}, abstract = {Bose-Einstein condensation of ultra cold atoms is typically realized in magnetic traps which effectively lead to an axially symmetric harmonic potential. This letter shows that the spectrum of collective vibrational modes of a repulsive condensate in a prolate potential displays a defect known as quantum monodromy. The monodromy is analysed on the basis of the dynamics of quasiparticles. In terms of the quasiparticles the regime of collective modes or the so-called hydrodynamic regime is characterized through kinetic energies much smaller than the chemical potential. In this limit the classical dynamics of the quasiparticles is integrable. The monodromy is quantitatively described by a monodromy matrix that is calculated from classical actions.}

}

@ARTICLE{Waalkens2001a,

AUTHOR = {H. Waalkens and C. Jung and H. S. Taylor}, TITLE = {Semiclassical assignment of the vibrational spectrum of {N}$_2${O}}, JOURNAL = {J. Phys. Chem. A}, VOLUME = {106}, PAGES = {911-924}, YEAR = 2002, URL = {http://dx.doi.org/10.1021/jp013057w}, key = {published}, abstract = {The vibrational spectrum of N2O as given by an effective spectroscopic Hamiltonian based on the existence of a superpolyad number is analyzed and assigned in terms of classical motions. The effective Hamiltonian includes a large number of resonances of which only one is dominant for low and intermediate superpolyad numbers. In this energy range, the corresponding classical system is quasi-integrable and can be described in terms of a system with only one nontrivial degree of freedom. This integrable system can be analyzed by considering the so-called “quantizing trajectories” on a “polyad sphere”. This method is no longer applicable when the superpolyad number is further increased and classical chaos comes into play. We then turn to a powerful universal method based on the graphical representation of semiclassical wave functions on a naturally appearing toroidal configuration space. These wave functions are obtained using the already known transformation matrix used in fitting the effective Hamiltonian. Experience with the interpretation of the resulting figures allows one to draw conclusions on the classical internal motions and therefore on the assignment of the quantum states without any further calculation. As such, the method is of particular interest to nontheorists and to nonspecialists in the fields of nonlinear dynamics and quantum calculation. For higher superpolyad numbers, the chaos remains mainly concentrated about the direct neighborhood of a separatrix of the former integrable system so that a great part of the vibrational spectrum can still be assigned in terms of the EBK quantum numbers of quantized tori.}

}

@ARTICLE{WD98,

AUTHOR = {H. Waalkens and H. R. Dullin}, TITLE = {Quantum Monodromy in Prolate Ellipsoidal Billiards}, JOURNAL = {Ann. Phys. (NY)}, VOLUME = {295}, PAGES = {81-112}, YEAR = 2002, key = {published}, abstract = {This is the third in a series of three papers on quantum billiards with elliptic and ellipsoidal boundaries. In the present paper we show that the integrable billiard inside a prolate ellipsoid has an isolated singular point in its bifurcation diagram and, therefore, exhibits classical and quantum monodromy. We derive the monodromy matrix from the requirement of smoothness for the action variables for zero angular momentum. The smoothing procedure is illustrated in terms of energy surfaces in action space including the corresponding smooth frequency map. The spectrum of the quantum billiard is computed numerically and the expected change in the basis of the lattice of quantum states is found. The monodromy is already present in the corresponding two-dimensional billiard map. However, the full three degrees of freedom billiard is considered as the system of greater relevance to physics. Therefore, the monodromy is discussed as a truly three-dimensional effect.}

}

@ARTICLE{DRVW99,

AUTHOR = {Dullin, H. R. and Richter, P. H. and Veselov, A. and Waalkens, H.}, TITLE = {Actions of the {N}eumann system via {P}icard-{F}uchs equations}, JOURNAL = {Physica D}, VOLUME = {155}, PAGES = {159-183}, YEAR = 2001, key = {published}, abstract = {The Neumann system describing the motion of a particle on an n-dimensional sphere with an anisotropic harmonic potential has been celebrated as one of the best understood integrable systems of classical mechanics. The present paper adds a detailed discussion and the determination of its action integrals, using differential equations rather than standard integral formulas. We show that the actions of the Neumann system satisfy a Picard–Fuchs equation which in suitable coordinates has a rather simple form for arbitrary n. We also present an explicit form of the related Gauß–Manin equations. These formulas are used for the numerical calculation of the actions of the Neumann system.}

}

@BOOK{Waalkens98,

AUTHOR = {H. Waalkens}, TITLE = {Elliptic and {E}llipsoidal {Q}uantum {B}illiards}, PUBLISHER = {Shaker Verlag}, ADDRESS = {Aachen}, YEAR = 1999, key = {published}, URL = {http://bookshop.blackwell.co.uk/bobuk/scripts/home.jsp?action=search&source=3266474136&type=isbn&term=3826566165}

}

@ARTICLE{WWD99,

AUTHOR = {H. Waalkens and J. Wiersig and H. R. Dullin}, TITLE = {Triaxial Ellipsoidal Quantum Billiards}, JOURNAL = {Ann. Phys. (NY)}, Volume = 276, Number = {1}, Pages = {64-110}, YEAR = 1999, key = {published}, abstract = {The classical mechanics, exact quantum mechanics and semiclassical quantum mechanics of the billiard in the triaxial ellipsoid are investigated. The system is separable in ellipsoidal coordinates. A smooth description of the motion is given in terms of a geodesic flow on a solid torus, which is a fourfold cover of the interior of the ellipsoid. Two crossing separatrices lead to four generic types of motion. The action variables of the system are integrals of a single Abelian differential of second kind on a hyperelliptic curve of genus 2. The classical separability carries over to quantum mechanics giving two versions of generalized Lamé equations according to the two sets of classical coordinates. The quantum eigenvalues define a lattice when transformed to classical action space. Away from the separatrix surfaces the lattice is given by EBK quantization rules for the four types of classical motion. The transition between the four lattices is described by a uniform semiclassical quantization scheme based on a WKB ansatz. The tunneling between tori is given by penetration integrals which again are integrals of the same Abelian differential that gives the classical action variables. It turns out that the quantum mechanics of ellipsoidal billiards is semiclassically most elegantly explained by the investigation of its hyperelliptic curve and the real and purely imaginary periods of a single Abelian differential.}

}

@ARTICLE{WWD97,

AUTHOR = {H. Waalkens and J. Wiersig and H. R. Dullin}, TITLE = {The Elliptic Quantum Billiard}, JOURNAL = {Ann. Phys. (NY)}, VOLUME = 260, NUMBER = {1}, PAGES = {50-90}, YEAR = 1997, key = {published}, abstract = {The exact and semiclassical quantum mechanics of the elliptic billiard is investigated. The classical system is integrable and exhibits a separatrix, dividing the phase space into regions of oscillatory and rotational motion. The classical separability carries over to quantum mechanics, and the Schrödinger equation is shown to be equivalent to the spheroidal wave equation. The quantum eigenvalues show a clear pattern when transformed into the classical action space. The implication of the separatrix on the wave functions is illustrated. A uniform WKB quantization taking into account complex orbits is shown to be adequate for the semiclassical quantization in the presence of a separatrix. The pattern of states in classical action space is nicely explained by this quantization procedure. We extract an effective Maslov phase varying smoothly on the energy surface, which is used to modify the Berry–Tabor trace formula, resulting in a summation over nonperiodic orbits. This modified trace formula produces the correct number of states, even close to the separatrix. The Fourier transform of the density of states is explained in terms of classical orbits, and the amplitude and form of the different kinds of peaks is analytically calculated.}

}

@ARTICLE{RDWW96,

AUTHOR = {P. H. Richter and H. R. Dullin and H. Waalkens and J. Wiersig}, TITLE = {Spherical Pendulum, Actions and Spin}, JOURNAL = {J. Phys. Chem.}, VOLUME = 100, PAGES = {19124-19135}, YEAR = {1996}, URL = {http://dx.doi.org/10.1021/jp9617128}, key = {published}, abstract = {The classical and quantum mechanics of a spherical pendulum are worked out, including the dynamics of a suspending frame with moment of inertia ?. The presence of two separatrices in the bifurcation diagram of the energy-momentum mapping has its mathematical expression in the hyperelliptic nature of the problem. Nevertheless, numerical computation allows to obtain the action variable representation of energy surfaces and to derive frequencies and winding ratios from there. The quantum mechanics is also best understood in terms of these actions. The limit ? ? 0 is of particular interest, both classically and quantum mechanically, as it generates two copies of the frameless standard spherical pendulum. This is suggested as a classical interpretation of spin.}

}

@TECHREPORT{DHJPSWWW97,

AUTHOR = {Dullin, H. R. and Heudecker, O. and Juhnke, M. and Pleteit, H. and Schwebler, H.-P. and Waalkens, H. and Wiersig, J. and Wittek, A.}, TITLE = {Energy Surfaces in Action Space}, TYPE = {Report}, INSTITUTION = {Institut f{\"u}r Dynamische Systeme}, SCHOOL = {Universit{\"a}t Bremen}, NUMBER = 406, key = {published}, YEAR = 1997

}

@MASTERSTHESIS{Waa95,

AUTHOR = {Waalkens, H.}, TITLE = {Semiklassische Berechnung chemischer Bindungen}, TYPE = {Diplomarbeit}, SCHOOL = {University of Oldenburg}, key = {published}, YEAR = 1995

}