Quantum three-body problems

A scheme for dealing with the quantum three-body problem is presented to separate the rotational degrees of freedom completely from the internal ones. In this method, the three-body Schrodinger equation is reduced to a system of coupled partial differential equations, depending only upon three internal variables. For arbitrary total orbital angular momentum / and the parity (− 1) ^l+λ (λ = 0 or 1), the number of the equations in this system isl = 1 −λ. By expanding the wavefunction with respect to a complete set of orthonormal basis functions, the system of equations is further reduced to a system of linear algebraic equations.     

Regions of prevalence in the coupled restricted three-body problems approximation

This work concerns the role played by a couple of the planar circular restricted three-body problem in the approximation of the bicircular model. The comparison between the differential equations governing the dynamics leads to the definition of region of prevalence where one restricted model provides the best approximation of the four-body model. According to this prevalence, the patched three-body problem approximation is used to design first guess trajectories for a spacecraft travelling under the Sun-Earth-Moon gravitational influence.     

Comparisons of integrators on a diverse collection of restricted three-body test problems

The restricted three-body problem has many important astronomicalapplications. We present a diverse collection of 28 symmetricand asymmetric, stable and unstable periodic orbits for thetwo- and three-dimensional problems and compare the performanceof eight general purpose integrators on the orbits.     

Quantum transfer matrices for discrete and continuous quasi-exactly solvable problems

We clarify the algebraic structure of continuous and discrete quasi-exactly solvable spectral problems by embedding them in the framework of the quantum inverse scattering method. The quasi-exactly solvable hamiltonians in one dimension are identified with traces of quantum monodromy matrices for specific integrable systems with nonperiodic boundary conditions. Applications to the Azbel-Hofstadter problem are outlined.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 8–24, July, 1995.     

Resonances in three-body scattering theory

Resonances of the three-body Schrödinger operator with exponentially decaying two-body interactions are characterized for negative energies as (1) poles of an analytically continued resolvent acting between certain exponentially weighted spaces; (2) eigenvalues of the Schrödinger operator acting in a suitable space; (3) singular points of the Faddeev matrix; (4) singular points of the Lippman—Schwinger operator; (5) poles of the S-matrix; (6) poles of analytic families of exponentially growing eigenfunctions.     

Intersecting invariant manifolds in spatial restricted three-body problems: Design and optimization of Earth-to-halo transfers in the Sun-Earth-Moon scenario

This work deals with the design of transfers connecting LEOs with halo orbits around libration points of the Earth-Moon CRTBP using impulsive maneuvers. Exploiting the coupled circular restricted three-body problem approximation, suitable first guess trajectories are derived detecting intersections between stable manifolds related to halo orbits of EM spatial CRTBP and Earth-escaping trajectories integrated in planar Sun-Earth CRTBP. The accuracy of the intersections in configuration space and the discontinuities in terms of Δv are controlled through the box covering structure implemented in the software GAIO. Finally first guess solutions are optimized in the bicircular four-body problem and single-impulse and two-impulse transfers are presented.     

Hill regions of charged three-body systems

For charged three-body systems, we discuss the configurations and orientations that are admissible for given values of the conserved total energy and angular momentum. The admissible configurations and orientations are discussed on a configuration space that is reduced by the translational, rotational and dilation symmetries of charged three-body systems. We consider the examples of the charged three-body systems given by the helium atom (two electrons and a nucleus) and the compound of two electrons and one positron. For comparison, the well known example of the Newtonian gravitational three-body system is discussed for the scheme presented in this paper first.     

Spurious terms in the three-body problem

The method of hyperharmonics is used to split the central two-body interactions and the Faddeev components of the wave function in a three-body system into physical and spurious terms. The sum of the physical terms of the interactions or of the Faddeev components for all pairs of particles is nonzero, whereas the sums of spurious terms of both the interactions and the Faddeev components over all pairs of particles vanish identically. We establish a criterion for the existence of spurious terms. We show that a sufficient condition for this criterion is equivalent to the conservation law for a certain quantum number. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 2, pp 253–271, November, 2000.     

Some homoclinic phenomena in the three-body problem

We prove the existence of transversal homoclinic points in the collinear three-body problem, restricted and general, and in the planar circular restricted three-body problem. As a consequence the shift of Bernoulli is proved to be included as a subsystem of a suitable section of the flow for the three cases studied. Then the existence of all the possible types of final evolution follows.     

Symbolic dynamics in the planar three-body problem

A chaotic invariant set is constructed for the planar three-body problem. The orbits in the invariant set exhibit many close approaches to triple collision and also excursions near infinity. The existence proof is based on finding appropriate “windows” in the phase space which are stretched across one another by flow-defined Poincaré maps.      

Periodic orbits of the restricted three-body problem

We prove, using a variational formulation, the existence of an infinity of periodic solutions of the restricted three-body problem. When the problem has some additional symmetry (in particular, in the autonomous case), we prove the existence of at least two periodic solutions of minimal period , for every . We also study the bifurcation problem in a neighborhood of each closed orbit of the autonomous restricted three-body problem.

     

From the restricted to the full three-body problem

The three-body problem with all the classical integrals fixed and all the symmetries removed is called the reduced three-body problem. We use the methods of symplectic scaling and reduction to show that the reduced planar or spatial three-body problem with one small mass is to the first approximation the product of the restricted three-body problem and a harmonic oscillator. This allows us to prove that many of the known results for the restricted problem have generalizations for the reduced three-body problem.

For example, all the non-degenerate periodic solutions, generic bifurcations, Hamiltonian-Hopf bifurcations, bridges and natural centers known to exist in the restricted problem can be continued into the reduced three-body problem. The classic normalization calculations of Deprit and Deprit-Bartholomé show that there are two-dimensional KAM invariant tori near the Lagrange point in the restricted problem. With the above result this proves that there are three-dimensional KAM invariant tori near the Lagrange point in the reduced three-body problem.