Wave-packet continuum discretization method for solving the three-body scattering problem

We generalize the wave-packet continuum discretization method previously developed for the scattering problem to the three-body system. For each asymptotic channel, we construct a basis of three-body wave packets given by square-integrable functions. We show that the projections of the channel resolvents on the subspace of three-body wave packets are determined by diagonal matrices, whose eigenvalues we find explicitly. We express the amplitudes of 2→2 processes explicitly in terms of “wave-packet” finite-dimensional projections of the full resolvent. To illustrate our formalism, we calculate the differential cross section of elastic deuteron scattering on a heavy nucleus above the three-body breakup threshold and the s-wave quartet (n-d)-scattering amplitude. The results of the calculations agree well with the results obtained by other methods. In terms of complexity, the proposed scheme for solving the three-body scattering problem is comparable to solving a similar problem for bound states. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 3, pp. 473–497, March, 2007.     

Asymptotic solution of a Coulomb multichannel scattering problem with a nonadiabatic channel coupling

Theoretical and Mathematical Physics - We develop a well-posed multiparticle Coulomb scattering problem based on asymptotic solutions of a Coulomb multichannel scattering problem constructed in the...     

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.     

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.     

Tridiagonal parametrization of interaction in the discrete approach to the scattering problem

We propose a method for constructing separable potentials with oscillatory formfactors given the scattering phase and the discrete spectrum of a quantum system. We show that the rank of the potential is minimum for an optimal value of the variational parameter ħω. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115. No. 2, pp. 263–274, May 1998     

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.      

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.     

Two-dimensional nuclear Coulomb scattering of a slow quantum particle

We study two-dimensional scattering of a quantum particle by the superposition of a Coulomb potential and a central short-range potential. We analyze the low-energy asymptotic behavior of all radial wave functions, partial phases, and scattering cross sections of such a particle. We propose two approaches for evaluating the scattering length and the effective radius.     

The discrete logarithm problem from a local duality perspective

The discrete logarithm problem is analyzed from the perspective of Tate local duality. Local duality in the multiplicative case and the case of Jacobians of curves over p-adic local fields are considered. When the local field contains the necessary roots of unity, the case of curves over local fields is polynomial time reducible to the multiplicative case, and the multiplicative case is polynomial time equivalent to computing discrete logarithm in finite fields. When the local field does not contains the necessary roots of unity, similar results can be obtained at the cost of going to an extension that contains these roots of unity. There was evidence in the analysis that suggests that the minimal extension where the local duality can be rationally and algorithmically defined must contain the roots of unity. Therefore, the discrete logarithm problem appears to be well protected against an attack using local duality. These results are also of independent interest for algorithmic study of arithmetic duality as they explicitly relate local duality in the case of curves over local fields to the multiplicative case and Tate-Lichtenbaum pairing (over finite fields).     

KAM Stability for a three-body problem of the Solar system

A new (iso-energetic) KAM method is tested on a specific three-body problem “extracted” from the Solar system (Sun-Jupiter + asteroid 12 Victoria). Analytical results in agreement with the observed data are established. This paper is a concise presentation of [2]. Supported by the MIUR projects: “Dynamical Systems: Classical, Quantum, Stochastic” and “Variational Methods and Nonlinear Differential Equations” Received: February 3, 2004