N-body theory revisited and its extension to the πNNN-NNN problem

In order to approach the pion-multinucleon problem, we have found it convenient to reformulate the generalN-body theory starting from the fully unclusterized (i.e.,NN) amplitude. If we rewrite such an amplitude in terms of new unknowns, which can be later identified as the amplitudes for all the (N–1)(N–1) cluster processes, and repeat recursively the procedure up to the treatment of the 22 cluster processes, we obtain very naturally the hierarchy of equations which ranges from theN-body fully-disconnected Lippmann-Schwinger equation to theN-body connected-kernel Yakubovskii-Grassberger-Sandhas one. This revisitation turns out to be very useful when considering the modifications required in case one of the bodies is a pion and the remaining are nucleons, with the pion being allowed to disappear and reappear through the action of a pion-nucleon vertex. In fact, we obtain a new set of coupled pionmultinucleon equations, which allow a consistent and simultaneous treatment of scattering and absorption. For the NNN system, the kernel of these coupled equations is shown to be connected after three iterations.Dedicated to Prof. Werner Sandhas on the occasion of his 60th birthday     

Universal Properties of the Four-Boson System in Two Dimensions

We consider the nonrelativistic four-boson system in two dimensions interacting via a short-range attractive potential. For a weakly attractive potential with one shallow two-body bound state with binding energy B₂, the binding energies B_N of shallow N-body bound states are universal and thus do not depend on the details of the interaction potential. We compute the four-body binding energies in an effective quantum mechanics approach. There are exactly two bound states: the ground state with B₄⁽⁰⁾=197.3(1)B₂ and one excited state with B₄⁽¹⁾=25.5(1)B₂. We compare our results to recent predictions for N-body bound states with large N1.On leave from FZ Jülich, Institut für Kernphysik (Theorie), D-52425 Jülich and HISKP (Theorie), Universität Bonn, Nußallee 14–16, D-53115 Bonn, Germany     

Coupling constant thresholds in nonrelativistic quantum mechanics

We extend the analysis of absorbtion of eigenvalues for the two body case to situations where absorbtion occurs at a two cluster threshold in anN-body system. The result depends on a Birman-Schwinger kernel for such anN-body system, an object which we apply in other ways. In particular, we control the number of discrete eigenvalues in the 0 limit.Research partially supported by U.S.N.S.F. under Grant MCS-78-01885.     

Many-body systems and the Efimov effect

We employ a Birman-Schwinger type analysis to derive estimates on the number of bound-states of certainN-body systems with threshold-energy =inf _ess(H) supposed to be zero. For many-body systems without any substructure we show that eigenvalues of the Schrödinger operatorH absorbed at =0 are in the point-spectrum ofH. Furthermore we characterize a multiparticle equivalent of the Efimov effect.     

Blow-up results of viriel type for Zakharov equations

We consider the Zakharov equations in ^N (for N=2,N=3). We first establish a viriel identity for such equations and then prove a blow-up result for solutions with a negative energy.     

On the discrete spectrum of the N-body quantum mechanical Hamiltonian: II

Using the Weinberg-van Winter equations we prove finiteness of the discrete spectrum of the N-body quantum mechanical Hamiltonian with pair potentials satisfying |V(x)| C(1 + |x|²)^– , > 1 in case the threshold of the continuous spectrum is negative and determined exclusively by eigenvalues of two-cluster Hamiltonians.     

Thermodynamics of theSU(N) Anderson impurity

The thermodynamic Bethe-ansatz equations of the degenerate Anderson model in theU limit with excluded multiple occupation of the localized level are solved numerically for the caseN=8. Thef-level occupation, the entropy, the spin and charge susceptibilities and the specific heat are obtained as a function of temperature for variousf-level energies. The results forN=6 andN=8 are compared with available data for CeTh and YbCuAl.     

N body scattering in the two-cluster region

We extend Combes' result on completeness ofN-body scattering at energies below the lowest 3-body threshold from potentials with |x|^–v– falloff ( number of dimensions for each particle) to central potentials with |x|^–1– . We also treat the scattering of electrons from neutral atoms in the two cluster region.Research supported by U.S. N.S.F. Grant MPS-75-11864     

N-body approaches to nuclear reactions

The transition from traditional nuclear reaction methods to more and more refined many-body approximations, which can be developed startirg from theN-body theory, is described in detail. We show how the intrinsic complicated structure of theN-body problem can be step-by-step incorporated so as to generate a sequence of approximations which approach the exact solution. Characteristic successive steps of this general approach are the bound-state approximation (as the lowest order of approximation), cluster models (as an intermediate stage) and well-founded approximation schemes embedded in the exact theory (as the final step).Presented at the symposium Mesons and Light Nuclei, Liblice, Czechoslovakia, June 1981.The author is pleased to thank Profs. C. Villi and G. Pisent for their encouraging interest in the subject of this work, and Dr. G. Cattapan for many stimulating discussions and useful suggestions.     

Ising model andN=2 supersymmetric theories

We establish a direct link between massive Ising model and arbitrary massiveN=2 supersymmetric QFT's in two dimensions. This explains why the equations which appear in the computation of spin-correlations in the non-critical Ising model are the same as those describing the geometry of vacua inN=2 theories. The tau-function appearing in the Ising model (i.e., the spin correlation function) is reinterpreted in theN=2 context as a new index. In special cases this new index is related to the Ray-Singer analytic torsion, and can be viewed as a generalization of that to the loop space of Kähler manifolds.     

Generalized Apollonian packings

In this paper we generalize the classical two-dimensional Apollonian packing of circles to the case where the circles are no more tangent. We introduce two elements ofSL(2,) as generators:R andT that are hyperbolic rotations of 2/3 and 2/N (N=2,3,4....), around two distinct points. The limit set of the discrete group generated byR andT provides, forN=7,8,9,.... a generalization of the Apollonian packing (which is itself recovered forN=). The valuesN=2,3,4,5 produce a very different result, giving rise to the rotation groups of the cube forN=2 and 4, and the icosahedron forN=3 and 5. ForN=6 the group is no longer discrete. To further analyze this structure forN7, we move to the Minkowski space in which the group acts on a one sheeted hyperboloid. The circles are now represented by points on this variety and generate a crystal on it.Laboratoire de l'Institut de Recherche Fondamentale du Commissariat à l'Energie Atomique     

Efimov Spectrum in Bosonic Systems with Increasing Number of Particles

It is well-known that three-boson systems show the Efimov effect when the two-body scattering length a is large with respect to the range of the two-body interaction. This effect is a manifestation of a discrete scaling invariance (DSI). In this work we study DSI in the N-body system by analysing the spectrum of N identical bosons obtained with a pairwise gaussian interaction close to the unitary limit. We consider different universal ratios such as \({E_N^0/E_3^0}\) and \({E_N^1/E_N^0}\) , with \({E_N^i}\) being the energy of the ground (i = 0) and first-excited (i = 1) state of the system, for \({N \leq16}\) . We discuss the extension of the Efimov radial law, derived by Efimov for N = 3, to general N.     

Large time behaviour of someN-body systems

In this work we prove completeness forN-body systems that evolve asymptotically into eitherN free particles or a two cluster system with one of the clusters being a single particle. For the three body case our results imply completeness for a very general system with potentials decaying like |x|^–1– at .This work was done at the Indian Statistical Institute, New Delhi 110016, India     

Solvable and/or Integrable and/or Linearizable N-Body Problems in Ordinary (Three-Dimensional) Space. I

Abstract

Several N -body problems in ordinary (3-dimensional) space are introduced which are characterized by Newtonian equations of motion (“acceleration equal force;” in most cases, the forces are velocity-dependent) and are amenable to exact treatment (“solvable” and/or “integrable” and/or “linearizable”). These equations of motion are always rotation-invariant, and sometimes translation-invariant as well. In many cases they are Hamiltonian, but the discussion of this aspect is postponed to a subsequent paper. We consider “few-body problems” (with, say, N =1,2,3,4,6,8,12,16,...) as well as “many-body problems” (N an arbitrary positive integer). The main focus of this paper is on various techniques to uncover such N -body problems. We do not discuss the detailed behavior of the solutions of all these problems, but we do identify several models whose motions are completely periodic or multiply periodic, and we exhibit in rather explicit form the solutions in some cases.     

Vector-spinor space and field equations

Generalizing the work of Einstein and Mayer, it is assumed that at each point of space-time there exists a vector-spinor space with N_v vector dimensions and N_s spinor dimensions, where N_v=2k and N_s=2 ^k, k3. This space is decomposed into a tangent space with4 vector and4 spinor dimensions and an internal space with N_v–4 vector and N_s–4 spinor dimension. A variational principle leads to field equations for geometric quantities which can be identified with physical fields such as the electromagnetic field, Yang-Mills gauge fields, and wave functions of bosons and fermions.     

On the discrete spectrum of theN-body quantum mechanical hamiltonian. I

We discussN-body kinematics and study the Berezin-Sigal equations in configuration space. Assuming that the threshold of the continuous spectrum is zero and that the pair potentials satisfy |V(x)|C(1+|x|²)^–Q , ³ > 1 (together with some technical hypotheses), we show that the discrete spectrum of the hamiltonian in the center of mass system is finite. The case of negative threshold will be treated in a further publication.This work was done while the author was staying at U.C. Berkeley and was partially supported by Pontificia Universidade Católica and by the Brazilian National Research Council (CNPq)     

N-dimensional classical integrable systems from Hopf algebras

A systematic method to constructN-body integrable systems is introduced by means of phase space realizations of universal enveloping Hopf algebras. A particular realization for theso(2, 1) case (Gaudin system) is analysed and an integrable quantum deformation is constructed by using quantum algebras as Poisson-Hopf symmetries.Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.