True Many-particle Scattering in the Oscillator Representation

We suggest an approach to many-particle scattering theory in the oscillator representation (OR) and treat the simplest case, in which the wave function in the asymptot ic domain has the form of a spherical wave in a multidimensional space. This corresponds to the so-called true many-particle scattering (TMS) [1, 2], i.e., to inclusion of only those states for which "democracy" is observed in the system and no single pair or group of particles is selected in the sense of forming bound states or scattering on an energy surface. Constructing a TMS wave function is part of the problem of finding a wave function for a many-body system [1]. However, investigating the TMS approximation is independently worthwhile from the physical s tandpoint because this approximation can adequately describe many processes in the disintegration of light nuclei into several fragments (see [3, 4]). Moreover, it ceases to be an approximation and becomes, in fact, an exact theory for the "democratic decay" processes [4], i.e., three-particle decay processes A -~ A1 + A2 + Aa in which no binary subsystem (AiAj) has a bound state, four-particle decay processes A --+ A1 + A2 + A3 + A4 in which no binary subsystem (AiAj) or trinary subsystem (AiAjAk) has a bound state, etc. Democratic decay processes have recently become very important because radioactive beams have begun to be used to systematically investigate exotic short-lived neutron-excessive nuclei, many of which can only decay over "democratic channels" (e.g., nL i --4 9Li + n + n, 6He --+ 4He + n + n, 14Be --+ 12Be + n + n, and the like) under low excitation energies that are of great interest at the present. The excited states of such nuclei have recently been theoretically investigated precisely in the framework of the TMS method (e.g., [5-10]). Some other currently central problems (e.g., resonances in twoand three-neutron systems [11]) are also studied in the TMS approximation. Furthermore, as is shown below, the TMS approximation can also be used to find the S-matrix for a many-particle system, and calculation of its poles not only can uniquely determine the parameters of resonance states but also can refine the bonding energy of the system. This refinement proves essential for weakly developed systems [8, 10]. The TMS approximation in this approach can probably be very exact for studying the ground state and other bound states of a many-particle system. It is natural to use the K-harmonics method, related to expanding the system wave function with respect to the hyperspherical basis, to calculate TMS characteristics (see [2-4] and references therein). We suggest that the expansion of the wave function with respect to the eigenfunctions of a (3A 3)-dimensional harmonic oscillator (A is the number of particles in the system) in the hyperspherical coordinates p and ~ be