Two-particle Subsystems Research Articles

Equivalence of relativistic three-particle quantization conditions

We show that a recently derived alternative form of the relativistic three-particle quantization condition for identical particles can be rewritten in terms of the R matrix introduced to give a unitary representation of the infinite-volume three-particle scattering amplitude. Combined with earlier work, this shows the equivalence of the relativistic effective field theory approach of Hansen and Sharpe (Refs.[1,2]) and the "finite-volume unitarity" approach of Mai and D\"oring (Refs.[3,4]). It also provides a generalization of the latter approach to arbitrary angular momenta of two-particle subsystems.

Efimov Effect for a Three-Particle System with Two Identical Fermions

We consider a three-particle quantum system in dimension three composed of two identical fermions of mass one and a different particle of mass $m$. The particles interact via two-body short range potentials. We assume that the Hamiltonians of all the two-particle subsystems do not have bound states with negative energy and, moreover, that the Hamiltonians of the two subsystems made of a fermion and the different particle have a zero-energy resonance. Under these conditions and for $m<m^* = (13.607)^{-1}$, we give a rigorous proof of the occurrence of the Efimov effect, i.e., the existence of infinitely many negative eigenvalues for the three-particle Hamiltonian $H$. More precisely, we prove that for $m>m^*$ the number of negative eigenvalues of $H$ is finite and for $m<m^*$ the number $N(z)$ of negative eigenvalues of $H$ below $z<0$ has the asymptotic behavior $N(z) \sim \mathcal C(m) |\log|z||$ for $z \rightarrow 0^-$. Moreover, we give an upper and a lower bound for the positive constant $\mathcal C(m)$.

Observation of an Efimov spectrum in an atomic system

In 1970, Vitaly Efimov predicted that three interacting particles can form an infinite series of bound trimer states, even when none of the two-particle subsystems is stable. Experimental evidence for such an exotic state was obtained in 2006, but now an Efimov spectrum, containing two such states with the predicted scaling between them, has been observed.

Theoretical description of two- and three-particle interactions in single ionization of helium by ion impact

Fil: Ciappina, Marcelo Fabian. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Bahia Blanca. Instituto de Fisica del Sur. Universidad Nacional del Sur. Departamento de Fisica. Instituto de Fisica del Sur; Argentina

Two-particle versus three-particle interactions in single ionization of helium by ion impact

We have performed kinematically complete experiments on single ionization of He by 100 MeV amu−1 C6+ and 3.6 MeV amu−1 Au24,53+ impact. By analysing doubly differential cross sections (DDCS) as a function of the momenta of all two-particle sub-systems we studied the importance of two-particle interactions. Furthermore, presenting the squared momenta of all three collision fragments simultaneously in a Dalitz plot, we evaluated the role of three-particle interactions. Finally, both for the DDCS and the Dalitz plots the corresponding correlation function was analysed. While the absolute cross sections confirm that ionization predominantly leads to a momentum exchange between the electron and the recoil-ion, the correlation function reveals strong correlations between the particles of any two-particle sub-system. Three-particle correlations, which are not accounted for by perturbative calculations, are quite significant as well, at least for certain kinematic conditions.

Essential and Discrete Spectra of the Three-Particle Schrödinger Operator on a Lattice

We consider the system of three quantum particles (two are bosons and the third is arbitrary) interacting by attractive pair contact potentials on a three-dimensional lattice. The essential spectrum is described. The existence of the Efimov effect is proved in the case where either two or three two-particle subsystems of the three-particle system have virtual levels at the left edge of the three-particle essential spectrum for zero total quasimomentum (K = 0). We also show that for small values of the total quasimomentum (K ≠ 0), the number of bound states is finite.

Relaxation of excited electrons in a paramagnetic electron gas: The role of spins in screening and scattering

The scattering lifetime of excited electrons close enough to the Fermi surface is investigated using the standard kinetic framework. This framework is implemented by effective, spin-dependent interactions and numerical phase-shift calculations. The nontrivial interplay between wave-mechanical interference effects in scattering and correlations in screening and dressing is discussed. Their role and relative importance are quantified via a comparative theoretical study. fermions ~not necessarily electrons, i.e., nucleons ! interact- ing through central forces is well-known, 7,8 The description is based on the free-space Schrodinger equation. We may suppose—and this will be the actual case in this work—that the spin-dependent part of the two-body effective interaction is proportional to the product s1is2 of two Pauli spin operators. 3,4,8 Generally, the scattering state function must be antisymmetric when all the coordinates of the two fermions ~spatial and spin! are interchanged. The rate of scattering is related 9 to the differential cross section ( ds) which is de- fined in terms ofproperly determined singlet and triplet am- plitudes. At these above circumstances the role of exchange appears as an interference effect in the differential cross section. 8 In the present work we shall investigate in detail the physical aspects of the effective interaction required in an implementation of the kinetic treatment of the relaxation process. 2 The influence of the fermion many-body environ- ment ''around'' our two-particle subsystem results in a short- range effective interaction between them. The scattering should be described, therefore, at least via an in-medium Schrodinger equation. The effective interaction must include electrostatic screening and exchange-correlation-mediated dressing, due to the Coulomb and Pauli correlations. These may affect the scattering characteristics ( ds) in a nontrivial manner in combination with the above-mentioned wave- interference effect. This paper is organized as follows. In Sec. II, we present the theoretical framework for the scattering lifetime, the models for the effective interaction, and the obtained results. These latter are given analytically and numerically in the form of illustrative figures. The results, based on different approximations, are discussed in a comparative way. The last section, Sec. III, is devoted to the conclusions. We shall use atomic units e 2 5\5me51 throughout this work.

Finiteness of the discrete spectrum of the Hamiltonian of a system of three arbitrary particles on a lattice

We prove that the number of bound states for the Hamiltonian of a system of three arbitrary particles interacting through pairwise attraction potentials on a three-dimensional lattice is finite in the cases where (1) none of the two-particle subsystems has a virtual level and (2) only one of the two-particle subsystems has a virtual level.

Finiteness of the discrete spectrum of the three-particle schrödinger equation on a lattice

Consideration is given to the Hamiltonian of a system of three identical quantum particles on a lattice that interact via pairwise contact attractive potentials. Finiteness of the three-particle bound states is proved for the three-dimensional discrete Schrodinger operator on the condition that the operators describing the two-particle subsystems have no virtual levels. For high dimensions (v ≥ 5), the finiteness of three-particle bound states is also proved in the presence of virtual levels.

Relativistic three-particle scattering equations.

We derive a set of relativistic three-particle scattering equations in the three-particle c.m. frame employing a relativistic three-particle propagator suggested long ago by Ahmadzadeh and Tjon in the c.m. frame of a two-particle subsystem. We make the coordinate transformation of this propagator from the c.m. frame of the two-particle subsystem to the three-particle c.m. frame. We also point out that some numerical applications of the Ahmadzadeh and Tjon propagator to the three-nucleon problem use unnecessary nonrelativistic approximations which do not simplify the computational task, but violate constraints of relativistic unitarity and/or covariance.

The Efimov effect. Discrete spectrum asymptotics

We study a three-particle Schrödinger operatorH for which none of the two-particle subsystems has negative bound states and at least two of them have zero energy resonances. We prove that under this condition the numberN(z) of bound states ofH belowz<0 has the asymptotics $$N(z) \sim \mathfrak{A}_0 |\log |z||$$ asz→-0, where the coefficient $$\mathfrak{A}_0 $$ depends only on the ratio of masses of the particles.

On the finiteness of the discrete spectrum of the Schr�dinger operator for a three-particle system with unstable two-particle subsystems in a homogeneous magnetic field

On the finiteness of the discrete spectrum of the Schr�dinger operator for a three-particle system with unstable two-particle subsystems in a homogeneous magnetic field

Spectral and scattering theory for 3-particle Hamiltonian with Stark effect : Non-existence of bound states and resolvent estimate

The present work is a continuation to [16], in which the author has proved the asymptotic completeness of wave operators for three-particle Stark Hamiltonians. In the proof there, the following two results about the spectral properties of two-particle subsystem Hamiltonians have played a central role: (1) non-existence of bound states; (2) uniform resolvent estimate at high energies. We here consider these two problems for three-particle systems and apply the obtained results to prove the asymptotic completeness for four-particle Stark Hamiltonians under the main assumption that any subsystem Hamiltonian does not have zero reduced charge.

On the finiteness of the discrete spectrum of Hamiltonians for quantum systems of three one- or two-dimensional particles

The system of three particles (acting in K dimensions (K=1, 2)) without stable two-particle subsystems is studied. For short-range potentials V ij (r ij ), the finiteness of the discrete spectrum is proved for any type of permutational symmetry.

'Partitions' in the perturbation theory of n-electron systems. III. The third-order energy

For pt.II, see ibid., vol.19, p.1863 (1986). In the perturbation theory of the general n-electron system, H(n)= Sigma nf(i)+ lambda Sigma ng(i,j), the subsystem method provides a finite decomposition (termed 'partition') in each order of the perturbation eigenfunctions and energies. The author shows that the third-order energy E(3)(n) is a weighted sum of third-order energies E(3)(n1) belonging to electronic states (most of them highly excited) of the n1=four-, three- and two-particle subsystems. The proof uses a parentage expansion of the second-order n-electron eigenfunction. The energy partitions can be regarded as the rigorous derivation of the important 'atomic energy relations' intuitively derived by Bacher and Goudsmit (1934).

Discretizing the three-body continuum: Bound states and two-body amplitudes

In this paper, we use a soluble model to investigate the accuracy of a discretization approximation for calculating bound states and two-body collision parameters. The model is that of three identical particles interacting via short-range, separable, S-wave pair potentials. Both weakly (spin-quartet fermion) and strongly (boson) interacting systems are examined. Discretization is achieved by diagonalizing the Hamiltonian for the two-particle subsystems in a finite set of L 2 functions, here chosen to be Laguerre polynomials. By requiring that the energy and eigenfunction of the lowest-lying member of the resulting set of pseudostates be in excellent agreement with the actual bound-state energy and eigenfunction of the two-particle Hamiltonian, this method discretizes the three-particle continuum, replacing it with a set of two-body continua corresponding to pseudo-inelastic scattering. Exact results have been obtained by solving the Alt-Grassberger-Sandhas (AGS) equations. Five sets of approximate results have also been determined. Four of these sets were obtained by applying the discretization method to four different three-particle collision formalisms, viz., those based on the AGS equations, the Lovelace equations, and the channel permuting array T ̂ and X equations. The fifth set was determined by solving the standard coupled reaction channel (CRC) equations augmented by adding the (excited) pseudostates to the ground state in each two-particle subsystem. The accuracy of the approximate results was studied as a function of the numbers of pseudostates included in the calculation; retaining only the two-particle ground state defines the bound-state approximation (BSA) for this system. In general, the BSA is better for the CRC method than for any of the discretized exact formalisms. In addition, the CRC method also responds best to the discretization approximation, e.g., its rate of convergence to the exact result is faster. Our explanation for this behavior is given in the succeeding article.

Integral equations for four strongly interacting particles, two of which are charged

The problem of the scattering in the systems of four strongly interacting particles with the Coulomb interaction is considered. By extracting and inverting in the kernels of the Faddeev-Yacubovsky equations the terms with the pure Coulomb two-particle transition operators, the Fredholm equations for the nuclear-Coulomb fourparticle transition operators are formulated. The kernels of thus regularized equations satisfy proper integral equations which are non-Fredholm in general case. The removal from these equations of the main Coulomb singularity arising owing to the interaction between the charges of two pairs of bound particles is performed. As it is known, the integral equations of motion describing the scattering in a fewparticle system become non-Fredholm in the presence of the Coulomb interaction [I] and must be reformulated to be applicable in practice. In the case of three particles a formal modification of the integral equations with the inclusion of the Coulomb potential in the non-perturbed Green function was proposed by Noble [2]. The approximate treatment of the Coulomb interaction in the Noble equations was considered in refs. [2, 3], the corresponding application to the proton-deuteron scattering was given in ref. [4]. The explicit separation from the kernel of the Faddeev equations for three particles of the dominant singularity, associated with the interaction between a charged particle and electric monopole moment of the two-particle subsystem, and the following inversion of it was carried out by Vesselova [5]. The relevant integral equations for three nucleons (p,d scattering) were derived in ref. [6]. The numerical solution of these equations was realized in [7]. A direct regularization of the three-particle integral equations by means of separation and inversion of all singularities being caused by the interaction of a charged particle with all multipoles of the two-particle subsystem was carried out in refs. [8, 9]. It was shown that the equations for the transition operator thus regularized were equivalent to the Noble equations obtained by the formal substitution of the Coulomb Green function for the free one. It was found in this way that the Nobletype equations were the result of the most complete regularization of the integral equations for three particles with Coulomb interaction. The method of direct regularization was found to be apt in the case of the system with the charge-exchange inter

Accuracy of the distorted-wave impulse approximation for breakup reactions in a three-body model

We have studied the reaction mechanisms of the ($p,2p$) reaction on $^{4}\mathrm{He}$ at energies of 65 and 100 MeV. The dynamics of the reaction is taken to be that of a three-particle system, the $^{4}\mathrm{He}$ target being viewed as a proton bound to an inert triton. The particles interact via separable potentials chosen to fit bound state and scattering data of the two-particle subsystems. The protons are treated as identical bosons interacting with each other only in the relative $S$ state while the proton-triton potential is limited to relative $L\ensuremath{\le}2$. The $^{4}\mathrm{He}(p,2p)^{3}\mathrm{H}$ doubly differential cross section can be calculated exactly within this model by solving the Faddeev equations numerically using the method of deformed contours and Pad\'e approximants. This can be compared with the distorted-wave impulse approximation (DWIA), calculated in this model by summing the appropriate subset of the multiple scattering series. This is done in momentum space and treats off-shell and finite range effects of the nucleon-nucleon $T$ matrix as well as recoil effects exactly. We find that the DWIA is an adequate approximation at 100 MeV but is almost double the exact result at 65 MeV. Its use in the extraction of spectroscopic information below 100 MeV is therefore suspect. The shapes of the exact, distorted, and plane wave cross sections agree quite well with each other (and with the experimental shape), differing only in magnitude.NUCLEAR REACTIONS $^{4}\mathrm{He}(p,2p)^{3}\mathrm{H}$, $E=65,100$ MeV; calculated $\ensuremath{\sigma}$. Threebody model, reaction mechanisms.

Field theory describing interacting two-particle subsystems

Field theory describing interacting two-particle subsystems

A field theory describing interacting two-particle subsystems

A field theory describing interacting two-particle subsystems