Diamagnetic Kepler Problem Research Articles

Fractional diamagnetic Kepler problem and elastic collisions

In this work, we consider an application of fractional derivatives to realistic physical situations, namely the elastic collision of particles and the nonintegrable diamagnetic Kepler problem. The origin of fractional dynamics can be nonlocal interacting dynamics, memory effects, environments with fractal interacting properties, and relaxation processes, among others. In the case of collisions, considering identical and distinguishable particles, additional solutions appear compared to non-fractional dynamics. For specific velocities of one particle before the collision, several velocities of the other particle are allowed after the collision. Consequently, novel velocity distributions emerge. For the diamagnetic Kepler problem, the fractional dynamic strongly affects the regular and quasi-regular regimes of motion, while the completely chaotic motion remains essentially unaltered. Besides, we derive a fractional momentum-like integral of motion for the pure fractional Kepler problem.

Electric-field effects on the closed orbits of the diamagnetic Kepler problem

Institute for Superconducting and Electronic Materials School of Physics University of Wollongong

Closed-orbit dependence on the field direction in the anisotropic diamagnetic Kepler problem

The classical orbits of an electron with anisotropic mass interacting with a Coulomb center in the presence of a magnetic field are investigated. It is shown that the shape and duration of the closed orbits depend on the magnetic-field direction. Reasonable agreement with available experimental results for donor-doped silicon samples in a magnetic field is found, when the energy value of the conduction electron is appropriately chosen.

Systematic Computer Assisted Proofs of periodic orbits of Hamiltonian systems

The numerical study of Dynamical Systems leads to obtain invariant objects of the systems such as periodic orbits, invariant tori, attractors and so on, that helps to the global understanding of the problem. In this paper we focus on the rigorous computation of periodic orbits and their distribution on the phase space, which configures the so called skeleton of the system. We use Computer Assisted Proof techniques to make a rigorous proof of the existence and the stability of families of periodic orbits in two-degrees of freedom Hamiltonian systems, which provide rigorous skeletons of periodic orbits. To that goal we show how to prove the existence and stability of a huge set of discrete initial conditions of periodic orbits, and later, how to prove the existence and stability of continuous families of periodic orbits. We illustrate the approach with two paradigmatic problems: the Hénon–Heiles Hamiltonian and the Diamagnetic Kepler problem.

Non-Hydrogenic Rydberg Atoms in Magnetic Fields

The classical and quantal dynamics of non-hydrogenic Rydberg atoms in magnetic fields are investigated. Previous attempts to infer classical behaviour from quantum properties produced conflicting results: at low scaled energies (ε=−0.5) the nearest-neighbour statistics (NNS) were found to be at the chaotic (Wigner) limit while quantum phase-space distributions suggested a high degree of regularity. Here the classical limit is investigated directly by solving the equations of motion of the Diamagnetic Kepler problem (DKP) with an additional non-Coulombic model potential. It is found that typically trajectories are, over a long time-scale, ergodic. However over a shorter time-scale—in between collisions with the core—classical trajectories remain confined on the tori of the DKP. The origin of a well-known resonance in the NNS of hydrogen at ε= −0.316 is clarified by the comparison with the non-hydrogenic behaviour. However, the classical model only partially explains the quantum behaviour. The difficulties of quantizing such a system are discussed.

Semiclassical quantization of the diamagnetic hydrogen atom with near–action-degenerate periodic-orbit bunches

The existence of periodic orbit bunches is proven for the diamagnetic Kepler problem. Members of each bunch are reconnected differently at self-encounters in phase space but have nearly equal classical action and stability parameters. Orbits can be grouped already on the level of the symbolic dynamics by application of appropriate reconnection rules to the symbolic code in the ternary alphabet. The periodic orbit bunches can significantly improve the efficiency of semiclassical quantization methods for classically chaotic systems, which suffer from the exponential proliferation of orbits. For the diamagnetic hydrogen atom the use of one or few representatives of a periodic orbit bunch in Gutzwiller's trace formula allows for the computation of semiclassical spectra with a classical data set reduced by up to a factor of 20.

Realization of Anisotropic Diamagnetic Kepler Problem in a Solid State Environment

The anisotropic diamagnetic Kepler problem (ADKP) is realized experimentally by the orbital electrons of a P donor in Si under magnetic fields. The interference of electron wave packets which leads to quasi-Landau resonances (QLR) were observed. Applying the closed-orbit theory to an anisotropic solid state environment, we have identified orbits responsible for the QLR manifesting the quantum chaotic behavior in Rydberg atoms. The excellent consistency between the measured spectra and theoretical calculation provides unambiguous evidence of quantum chaotic dynamics of electrons in the ADKP.

ROTATION NUMBERS OF INVARIANT MANIFOLDS AROUND UNSTABLE PERIODIC ORBITS FOR THE DIAMAGNETIC KEPLER PROBLEM

In this paper, a method to construct topological template in terms of symbolic dynamics for the diamagnetic Kepler problem is proposed. To confirm the topological template, rotation numbers of invariant manifolds around unstable periodic orbits in a phase space are taken as an object of comparison. The rotation numbers are determined from the definition and connected with symbolic sequences encoding the periodic orbits in a reduced Poincaré section. Only symbolic codes with inverse ordering in the forward mapping can contribute to the rotation of invariant manifolds around the periodic orbits. By using symbolic ordering, the reduced Poincaré section is constricted along stable manifolds and a topological template, which preserves the ordering of forward sequences and can be used to extract the rotation numbers, is established. The rotation numbers computed from the topological template are the same as those computed from their original definition.

The measure of chaoticity in stationary quantum systems

Our earlier suggested quantum chaoticity measure characterizing the initial symmetry breaking and the destruction of the corresponding integrals of motion in a perturbed system is used for the analysis of the quantum Henon-Heiles Hamiltonian and diamagnetic Kepler problem. We demonstrate that the critical perturbation parameter for the transition from regularity to chaos in the quantum systems is in remarkable agreement with the classical picture of chaotization. We also show that this critical parameter can be estimated in the framework of first-order perturbation theory.

A Method to Find Unstable Periodic Orbits for the Diamagnetic Kepler Problem

A method to determine the admissibility of symbolic sequences and to find the unstable periodic orbits corresponding to allowed symbolic sequences for the diamagnetic Kepler problem is proposed by using the ordering of stable and unstable manifolds. By investigating the unstable periodic orbits up to length 6, a one to one correspondence between the unstable periodic orbits and their corresponding symbolic sequences is shown under the system symmetry decomposition.

Ghost Orbit Bifurcations in Semiclassical Spectra

Gutzwiller's semiclassical trace formula for the density of states in a chaotic system diverges near bifurcations of periodic orbits, where it must be replaced with uniform approximations. It is well known that, when applying these approximations, complex predecessors of orbits created in the bifurcation (``ghost orbits'') can produce clear signatures in the semiclassical spectra. We demonstrate that these orbits themselves can undergo bifurcations, resulting in complex, non-generic bifurcation scenarios. We do so by studying an example taken from the Diamagnetic Kepler Problem. By application of normal form theory, we construct an analytic description of the complete bifurcation scenario, which is then used to calculate the pertinent uniform approximation. The ghost orbit bifurcation turns out to produce signatures in the semiclassical spectrum in much the same way as a bifurcation of real orbits would.

Ergodic Properties and Thermodynamics in Anharmonic Oscillator Systems**The project supported in part by National Natural Science Foundation of China, National Basic Research Project “Nonlinear Science”, the Research Grant Council RGC of Hong Kong and the Hong Kong Baptist University Faculty Research Grant FRG

Ergodic properties of anharmonic oscillator systems with only a few degrees of freedom are both theoretically and numerically investigated. Statistical mechanics and thermodynamics of ergodic Hamiltonian systems are built. The Henon–Heiles systems with reflecting walls and the diamagnetic Kepler problem are studied

Uniform Approximations for Non-generic Bifurcation Scenarios Including Bifurcations of Ghost Orbits

Gutzwiller's trace formula allows interpretation of the density of states of a classically chaotic quantum system in terms of classical periodic orbits. It diverges when periodic orbits undergo bifurcations and must be replaced with a uniform approximation in the vicinity of the bifurcations. As a characteristic feature, these approximations require the inclusion of complex “ghost orbits”. By studying an example taken from the Diamagnetic Kepler Problem, viz. the period-quadrupling of the balloon orbit, we demonstrate that these ghost orbits themselves can undergo bifurcations, giving rise to non-generic complicated bifurcation scenarios. We extend classical normal form theory so as to yield analytic descriptions of both bifurcations of real orbits and ghost orbit bifurcations. We then show how the normal form serves to obtain a uniform approximation taking the ghost orbit bifurcation into account. We find that the ghost bifurcation produces signatures in the semiclassical spectrum in much the same way as a bifurcation of real orbits does.

Symbolic Dynamics of the Diamagnetic Kepler Problem WithoutInvolving Bounces

Without involving bounce events, a Poincaré sectionassociated with the axes is found to give a map on the annulus for thediamagnetic Kepler problem. Symbolic dynamics is then establishedbased on the lift of the annulus map. The correspondence between thecoding derived from this axis Poincaré section is compared withthe coding based on bounces. Symmetry is used to reduce the symbolicdynamics. By means of symbolic dynamics the admissibility of periodicorbits is analyzed, and the symmetry of orbitsdiscussed.

Significance of ghost orbit bifurcations in semiclassical spectra

Gutzwiller's trace formula for the semiclassical density of states in a chaotic system diverges near bifurcations of periodic orbits, where it must be replaced with uniform approximations. It is well known that, when applying these approximations, complex predecessors of orbits created in the bifurcation (`ghost orbits') can produce pronounced signatures in the semiclassical spectra in the vicinity of the bifurcation. It is the purpose of this paper to demonstrate that these ghost orbits can also undergo bifurcations, resulting in complex, nongeneric bifurcation scenarios. We do so by studying an example taken from the diamagnetic Kepler problem, namely the period quadrupling of the balloon orbit. By application of normal form theory we construct an analytic description of the complete bifurcation scenario, which is then used to calculate the pertinent uniform approximation. The ghost orbit bifurcation turns out to produce signatures in the semiclassical spectrum in much the same way as a bifurcation of real orbits would.

Uniform semiclassical approximations for umbilic bifurcation catastrophes

Gutzwiller's trace formula for the semiclassical density of states diverges at the bifurcation points of periodic orbits and has to be replaced with uniform semiclassical approximations. We present a method to derive these expressions from the standard representations of the elementary catastrophes and to directly relate the uniform solutions to classical periodic orbit parameters, thereby circumventing the numerical application of normal form theory. The technique allows an easy handling of ungeneric bifurcations with corank 2 such as the umbilic catastrophes and is demonstrated on a hyperbolic umbilic in the diamagnetic Kepler problem.

The semiclassical resonance spectrum of hydrogen in a constant magnetic field

We present the first purely semiclassical calculation of the resonance spectrum in the diamagnetic Kepler problem (DKP), a hydrogen atom in a constant magnetic field with . The classical system is unbound and completely chaotic for a scaled energy larger than a critical value . The quantum mechanical resonances can in semiclassical approximation be expressed as the zeros of the semiclassical zeta function, a product over all the periodic orbits of the underlying classical dynamics. Intermittency originating from the asymptotically separable limit of the potential at large electron - nucleus distance causes divergences in the periodic orbit formula. Using a regularization technique introduced in (Tanner G and Wintgen D 1995 Phys. Rev. Lett. 75 2928) together with a modified cycle expansion, we calculate semiclassical resonances, both position and width, which are in good agreement with quantum mechanical results obtained by the method of complex rotation. The method also provides good estimates for the bound state spectrum obtained here from the classical dynamics of a scattering system. A quasi-Einstein - Brillouin - Keller (QEBK) quantization is derived that allows for a description of the spectrum in terms of approximate quantum numbers and yields the correct asymptotic behaviour of the Rydberg-like series converging towards the different Landau thresholds.

Bifurcations of electronic Rydberg wavepackets and core scattering

The influence of core scattering effects on the dynamics of a bifurcating electronic Rydberg wavepacket is investigated in the diamagnetic Kepler problem. Results are presented for a spherically symmetric nonhydrogenic core with particular emphasis on nongeneric bifurcations of the Edmonds - Garton - Tomkins orbit.

Bifurcation of the periodic orbits of Hamiltonian systems: An analysis using normal form theory.

We develop an analytic technique to study the dynamics in the neighborhood of a periodic trajectory of a Hamiltonian system. The theory begins with Poincar\'e and Birkhoff; major modern contributions are due to Meyer, Arnol'd, and Deprit. The realization of the method relies on local Fourier-Taylor series expansions with numerically obtained coefficients. The procedure and machinery are presented in detail on the example of the ``perpendicular'' (z=0) periodic trajectory of the diamagnetic Kepler problem. This simple one-parameter problem well exhibits the power of our technique. Thus, we obtain a precise analytic description of bifurcations observed by J.-M. Mao and J. B. Delos [Phys. Rev. A 45, 1746 (1992)] and explain the underlying dynamics and symmetries. \textcopyright{} 1996 The American Physical Society.

Alternative method to find orbits in chaotic systems.

We present here a method which applies well ordered symbolic dynamics to find unstable periodic and nonperiodic orbits in a chaotic system. The method is simple and efficient and has been successfully applied to a number of different systems such as the H\'enon map, disk billiards, stadium billiard, wedge billiard, diamagnetic Kepler problem, collinear Helium atom, and systems with attracting potentials. The method seems to be better than earlier applied methods.