Optimized Lower Bound for Four-Body Hamiltonians

Gauge symmetry, chirality and parity effects in four-particle systems: Coulomb's law as a universal function for diatomic molecules

“…Nevertheless, it gives us a hint for a universal potential fitting in a Coulomb model with gauge symmetry, in a form ± (1-1/m) we wanted and roughly in accordance with observation, see When t = 1, the Hamiltonian (14a) can directly be rewritten, if the small perturbation is neglected, as H XY = H X-+ H Y+ -e 2 /R 12 -(IE X + EA X )+ 0 -e 2 /R 12 (14e) the simple one-term generic Coulomb solution, nearly conforming to (9a) or (11). We first discuss some implications of (14e) and of intra atomic charge inversion in the Hamiltonian.…”

Bounding the ground-state energy of a many-body system with the differential method

“…However, if one has some experimental clues about ǫ ij (by measuring 2-body masses or dissociation energies) or numerical estimates as well, it is always interesting to obtain some relations between the ǫ's and the ground-state energies of larger systems. As mentioned in the introduction, this have been achieved in [2] for N = 3 and in [4] for N = 4 when the interactions are of the form v ij (r) ∝ sign(β)r β . For β > 0, the semiclassical argument given at the end of section 2 shows that (13) is expected to be unbounded; then (4) gives no information.…”

“…As mentioned in the introduction, this have been achieved in [2] for N = 3 and in [4] for N = 4 when the interactions are of the form v ij (r) ∝ sign(β)r β . For β > 0, the semiclassical argument given at the end of section 2 shows that (13) is expected to be unbounded; then (4) gives no information. If we want to take the advantage of the simple form (13) (that is, to keep the choice (9) with (11) for the test functions), we have to work with finite range potentials.…”

“…The trial wave-function (9) of the global system must be kept square-integrable but it is not necessary for all two-body subsystems to have a bound state when isolated 5 . For instance, for two electric charges 4 The present paper wants mainly to stress the simplicity of the differential method. It does not seek for a real performance at the moment and we will not try to improve the choice of coordinates.…”

“…Such an approach has been successfully applied to Coulombian (bosonic and fermionic) systems of charged particles [12] or self-gravitating bosons [13,1]. Clever refinements have been proposed that provide some very accurate lower bounds of E 0 for the threebody [3] and the four-body systems [4]. Though not easily generalizable to an arbitrary number of particles, these last optimized variational methods can be applied to interactions that are not necessarily Coulombian and may be relevant for quarks models [2] where some inequalities between baryon and meson masses represent theoretical, numerical and experimental substantial information [17,20, and references therein].…”