Equations In Configuration Space Research Articles

The charge and mass symmetry breaking in the KK K¯ system

In the framework of the Faddeev equations in configuration space, we investigate the K(1460) meson as a resonant state of the KKK¯ kaonic system. We perform calculations for the particle configurations K 0 K + K − and K0K+K0¯ within two models: the ABC model, in which all three particles are distinguishable, and the AAC model when two particles are identical. The models differ in their treatment of the kaon mass difference and the attractive Coulomb force between the K + K − pair. We found that the Coulomb shift adds over 1 MeV to the three-body binding energy. The expected correction to the binding energy due to mass redistribution from AA to AB is found to be negligible, up to a maximum of 6% of the relative mass correction. At the same time, the symmetry of the wave function is distorted depending on the mass ratio value. We found that the repulsive KK interaction plays an essential role in the binding energy of the KKK¯ system and reports the mass of 1461.8 or 1464.1 MeV for the neutral K 0(1460) and 1466.5 or 1468.8 MeV for the charged K +(1460) resonances, respectively, depending on the parameter sets for KK and KK¯ interactions.

Semiclassical theory and the Koopman-van Hove equation * *Contribution to the Special Collection ‘Koopman Methods in Classical and Quantum-Classical Mechanics’.

The phase space Koopman-van Hove (KvH) equation can be derived from the asymptotic semiclassical analysis of partial differential equations. Semiclassical theory yields the Hamilton–Jacobi equation for the complex phase factor and the transport equation for the amplitude. These two equations can be combined to form a nonlinear semiclassical version of the KvH equation in configuration space. There is a natural injection of configuration space solutions into phase space and a natural projection of phase space solutions onto configuration space. Hence, every solution of the configuration space KvH equation satisfies both the semiclassical phase space KvH equation and the Hamilton–Jacobi constraint. For configuration space solutions, this constraint resolves the paradox that there are two different conserved densities in phase space. For integrable systems, the KvH spectrum is the Cartesian product of a classical and a semiclassical spectrum. If the classical spectrum is eliminated, then, with the correct choice of Jeffreys–Wentzel–Kramers–Brillouin (JWKB) matching conditions, the semiclassical spectrum satisfies the Einstein–Brillouin–Keller quantization conditions which include the correction due to the Maslov index. However, semiclassical analysis uses different choices for boundary conditions, continuity requirements, and the domain of definition. For example, use of the complex JWKB method allows for the treatment of tunneling through the complexification of phase space. Finally, although KvH wavefunctions include the possibility of interference effects, interference is not observable when all observables are approximated as local operators on phase space. Observing interference effects requires consideration of nonlocal operations, e.g. through higher orders in the asymptotic theory.

Emergence of 4H Jπ = 1− resonance in contact theories

We obtain the s- and p-wave low-energy scattering parameters for n3H elastic scattering and the position of the 4H Jπ=1− resonance using the pionless effective field theory at leading order. Results are extracted with three numerical techniques: confining the system in a harmonic oscillator trap, solving the Faddeev-Yakubovsky equations in configuration space, and using an effective two-body cluster approach. The renormalization of the theory for the relevant amplitudes is assessed in a cutoff-regulator range between 1fm−1 and 10fm−1.Most remarkably, we find a cutoff-stable/RG-invariant resonance in the 4H Jπ=1− system. This p-wave resonance is a universal consequence of a shallow two-body state and the introduction of a three-body s-wave scale set by the triton binding energy. The stabilization of a resonant state in a few-fermion system through pure contact interactions has a significant consequence for the powercounting of the pionless theory. Specifically, it suggests the appearance of similar resonant states also in larger nuclei, like 16-oxygen, in which the theory's leading order does not predict stable states. Those resonances would provide a starting state to be moved to the correct physical position by the perturbative insertion of sub-leading orders, possibly resolving the discrepancy between data and contact EFT.

Probabilistic Collision-free Pattern Control For Large-Scale Spacecraft Swarms Around Circular Orbits

Abstract This work considers controlling large-scale spacecraft swarms to achieve complex spatial configuration. A novel distributed guidance algorithm is proposed based on Inhomogeneous Markov Chains, Probabilistic Density Guidance and Voronoi partition (IMC-PDG-Voronoi) algorithms. The physical space is partitioned into multiple bins and the density distribution of the swarm is controlled via a probabilistic approach. Then the modified Voronoi partition method is used to generate a collision-free trajectory for each agent. To apply the probabilistic control algorithm to circular Earth orbit, the periodic solution of the Clohessy-Wiltshire (C-W) equation in configuration space is transformed into a parameter space. Then a convex optimization open-loop controller with minimum fuel consumption in LVLH coordinates is designed to control the swarm to expected positions. Numerical simulations show that the algorithm can effectively guide and control large-scale spacecraft swarms to form complex configurations on circular orbits, with high precision and little cost.

3N potentials in the Faddeev coordinate space approach to Nd scattering

For studying 3N bound states and Nd scattering, the Tucson Melbourne (TM) and Urbana 3N forces have been derived from the chiral EFT in the momentum representation. The Faddeev equations in configuration space have attractive properties when applied for nd and pd scattering above the twobody threshold. For that reason, we derived components of the TM 3N potential in the coordinate representation.

Three-body model for K(1460) resonance

The three-body $KK\bar K$ model for the $K(1460)$ resonance is developed on the basis of the Faddeev equations in configuration space. A single-channel approach is using with taking into account the difference of masses of neutral and charged kaons. It is demonstrated that a splitting the mass of the $K(1460)$ resonance takes a place around 1460 MeV according to $K^0K^0{\bar K}^0$, $K^0K^+K^-$ and $K^+K^0{\bar K}^0$, $ K^+K^+K^-$ neutral and charged particle configurations, respectively. The calculations are performed with two sets of $KK$ and $K\bar K$ phenomenological potentials, where the latter interaction is considered the same for the isospin singlet and triplet states. The effect of repulsion of the $KK$ interaction on the mass of the $KK\bar K$ system is studied and the effect of the mass polarization is evaluated. The first time the Coulomb interaction for description of the $K(1460)$ resonance is considered. The mass splitting in the $K$(1460) resonances is evaluated to be in range of 10 MeV with taking into account the Coulomb force. The three-body model with the $K\bar K$ potential, which has the different strength of the isospin singlet and triplet parts that are related by the condition of obtaining a quasi-bound three-body state is also considered. Our results are in reasonable agreement with the experimental mass of the $K(1460)$ resonance.

Space-filter techniques for quasi-neutral hybrid-kinetic models

The space-filter approach has been proved as a fundamental tool in studying turbulence in neutral fluids, providing the ability to analyze scale-to-scale energy transfer in configuration space. It is well known that turbulence in plasma presents challenges different from neutral fluids, especially when the scale of interest includes kinetic effects. The space-filter approach is still largely unexplored for kinetic plasma. Here, we derive the space-filtered (or equivalently “coarse-grained”) equations in configuration space for a quasi-neutral hybrid-kinetic plasma model, in which ions are fully kinetic and electrons are a neutralizing fluid. Different models and closures for the electron fluid are considered, including finite electron-inertia effects and full electrons' pressure-tensor dynamics. Implications for the cascade of turbulent fluctuations in real space depending on different approximations are discussed.

Description of Four- and Five-Nucleon Systems by Solving Faddeev-Yakubovsky Equations in Configuration Space

The Faddeev Yakubovsky equations constitute a rigorous formulation of the quantum mechanical N body problem in the framework of non relativistic dynamics. They allow the exact solutions of the Schrodinger equation for bound and scattering states to be obtained. In this review, we will present the general formalism as well as the numerical tools we use to solve Faddeev Yakubovsky equations in configuration space. We will consider in detail the description of the four and five nucleon systems based on modern realistic nuclear Hamiltonians. Recent achievements in this domain will be summarized. Some of the still controversial issues related with the nuclear Hamiltonians as well as the numerical methods traditionally employed to solve few nucleon problems will be highlighted.

The Faddeev–Yakubovsky Symphony

We briefly summarize the main steps leading to the Faddeev-Yakubovsky equations in configuration space for N=3, 4 and 5 interacting particles.

Ab initio calculations of 5H resonant states

By solving the 5-body Faddeev-Yakubovsky equations in configuration space with realistic nuclear Hamiltonians we have studied the resonant states of 5H isotope. Two different methods, allowing to bypass the exponentially diverging boundary conditions, have been employed providing consistent results. The existence of 5H broad Jπ=1/2+,3/2+,5/2+ states as S-matrix poles has been confirmed and compared with the, also calculated, resonant states in 4H isotope. We have established that the positions of these resonances only mildly depend on the nuclear interaction model.

On Mass Polarization Effect in Three-Body Nuclear Systems

We evaluate the mass polarization term of the kinetic-energy operator for different three-body nuclear $AAB$ systems by employing the method of Faddeev equations in configuration space. For a three-boson system this term is determined by the difference of the doubled binding energy of the $AB$ subsystem $2E_{2}$ and the three-body binding energy $E_{3}(V_{AA}=0)$ when the interaction between the identical particles is omitted. In this case: $\left\vert E_{3}(V_{AA}=0)\right\vert >2\left\vert E_{2}\right\vert$. In the case of a system complicated by isospins(spins), such as the kaonic clusters $ K^{-}K^{-}p$ and $ppK^{-}$, the similar evaluation impossible. For these systems it is found that $\left\vert E_{3}(V_{AA}=0)\right\vert <2\left\vert E_{2}\right\vert$. A model with an $AB$ potential averaged over spin(isospin) variables transforms the later case to the first one. The mass polarization effect calculated within this model is essential for the kaonic clusters. Besides we have obtained the relation $|E_3|\le |2E_2|$ for the binding energy of the kaonic clusters.

Description of Five-Nucleon Systems Using Faddeev-Yakubovsky Equations

A brief overview of Faddeev-Yakubovsky equations is presented before deriving 5-body ones. Numerical formalism, enabling to solve these equations in configuration space for a system of five nucleons is described. Microscopic calculations are realized to determine phaseshifts of low energy neutron scattering on $$^4$$ He and $$1/2^+$$ resonance position of $$^5$$ H, employing phenomenological MT I-III potential.

On binding energy of trions in bulk materials

We study the negatively T− and positively T+ charged trions in bulk materials in the effective mass approximation within the framework of a potential model. The binding energies of trions in various semiconductors are calculated by employing Faddeev equation in configuration space. Results of calculations of the binding energies for T− are consistent with previous computational studies and are in reasonable agreement with experimental measurements, while the T+ is unbound for all considered cases. The mechanism of formation of the binding energy of trions is analyzed by comparing contributions of a mass-polarization term related to kinetic energy operators and a term related to the Coulomb repulsion of identical particles.

Weakly Bound Li He $$_\mathbf{2 }$$ 2 Molecules

The results for binding energies of $^6$Li He$_2$ and $^7$Li He$_2$ systems are presented. They are obtained by solving Faddeev equations in configuration space. It is shown that the excited states in both systems are of the Efimov-type.

Time reversal invariance violating and parity conserving effects in proton-deuteron scattering

Time reversal invariance violating and parity conserving effects for low-energy elastic neutron-deuteron scattering are calculated for meson exchange and effective field theory type potentials in a distorted wave-born approximation using realistic hadronic wave functions, obtained by solving three-body Faddeev equations in configuration space.

Three-body calculations for the K−pp system within potential models

We present three-body nonrelativistic calculations within the framework of a potential model for the kaonic cluster K−pp using two methods: the method of hyperspherical harmonics in the momentum representation and the method of Faddeev equations in configuration space. To perform numerical calculations, different NN and antikaon–nucleon interactions are applied. The results of the calculations for the ground-state energy for the K−pp system obtained by both methods are in reasonable agreement. Although the ground-state energy is not sensitive to the pp interaction, it shows very strong dependence on the K−p potential. We show that the dominant clustering of the system in the configuration Λ (1405) + p allows us to calculate the binding energy to good accuracy within a simple cluster approach for the differential Faddeev equations. The theoretical discrepancies in the binding energy and width for the K−pp system related to the different pp and K−p interactions are addressed.

Erratum: Time reversal invariance violating and parity conserving effects in neutron-deuteron scattering [Phys. Rev. C84, 025501 (2011)

Time reversal invariance violating parity conserving (TVPC) effects are calculated for elastic proton deuteron scattering with proton energies up to $2~$MeV. Distorted Wave Born Approximation is employed to estimate TVPC matrix elements, based on hadronic wave functions, obtained by solving three-body Faddeev-Merkuriev equations in configuration space with realistic potentials.

Asymptotics of the binary amplitude for a model Faddeev equation

A model equation obtained from the s-wave Faddeev equation in configuration space for three identical bosons by replacing the inhomogeneous integral term with a known function is studied. This function simulates the asymptotic decrease of the inhomogeneous term in the original Faddeev equation when y → ∞ as ∼y–3/2. The asymptotes of the amplitude functions that approach the scattering amplitudes when y → ∞ are obtained analytically for the model equation using the Green function. It is shown that the asymptotics of the binary channel amplitude function oscillates. The similar oscillating behavior of the binary amplitude function is numerically demonstrated for the original s-wave Faddeev equation describing the neutron–deuteron scattering process.

The general configuration-space Faddeev formalism for studying pd scattering

The configuration-space Faddeev equations are derived for p-d scattering taking into account the difference in interaction between the participant particles. Appropriate modifications have been made in the well-known configuration-space equations for n-d scattering. To show the effect of these modifications, the s-wave calculations are performed for bound state and scattering problems. We model the charge symmetry breaking effect for 3 H and 3 He with a modified Malfliet-Tjon MT I-III potential. Results obtained for elastic n-d and p-d scattering at Elab =14.1 MeV are compared with our prediction (Ref. [1]) and those of the Los-Alamos/Iowa group (Ref. [2]) .

A Global Simulation Method for Flexible Multibody Systems With Variable Topology Structures

By means of a recursive formulation method, a generalized impulse–momentum-balance method, and a constraint violation elimination (CVE) method, we propose a new global simulation method for flexible multibody systems with kinematic structure changes. The constraint equations of a pair of adjacent bodies, considering body flexibility in Cartesian space, are derived for a recursive formulation. Constraint equations in configuration space, which are obtained from the constraints presented in this paper via recursive formulation, are very useful for modeling different kinematic structures and impacting governing equations. The novelty is that the impact governing equations, which calculate the jumps of generalized velocities, are modified by taking velocity-level CVE into consideration. Numerical examples are given to validate the presented method. Simulation results show that the new method can effectively suppress constraint drifts at the velocity level and stabilize constraint violations at the position level.