Variational, multibox approach is proposed to construct extremely accurate, bound-state wave functions for arbitrary three-body systems. The high efficiency of our present approach is based on an optimal choice of nonlinear parameters in the exponential basis functions. The proposed method is very flexible, since the final wave function can also include a large number of separately optimized cluster fragments. The wave functions obtained are very compact and highly accurate. Such wave functions can be used to compute various bound state properties for different three-body systems. The proposed approach has been successfully tested on a large number of actual systems. It is shown that the present approach can be used to solve various three-body problems with, in principle, arbitrary precision. In particular, the long-standing problem of highly accurate determination of the weakly bound (1,1) states in the ddmu and dtmu muonic molecular ions has finally been solved. The determined binding energies are -1.974 988 088 0+/-5 x 10(-10) eV and -0.660 338 74+/-1 x 10(-8) eV, respectively.

The Hamiltonian formulation of the tetrad gravity in any dimension higher than two, using its first order form when tetrads and spin connections are treated as independent variables, is discussed and the complete solution of the three dimensional case is given. For the first time, applying the methods of constrained dynamics, the Hamiltonian and constraints are explicitly derived and the algebra of the Poisson brackets among all constraints is calculated. The algebra of the Poisson brackets among first class secondary constraints locally coincides with Lie algebra of the ISO(2,1) Poincaré group. All the first class constraints of this formulation, according to the Dirac conjecture and using the Castellani procedure, allow us to unambiguously derive the generator of gauge transformations and find the gauge transformations of the tetrads and spin connections which turn out to be the same found by Witten without recourse to the Hamiltonian methods [Nucl. Phys. B 311 (1988) 46 ]. The gauge symmetry of the tetrad gravity generated by Lie algebra of constraints is compared with another invariance, diffeomorphism. Some conclusions about the Hamiltonian formulation in higher dimensions are briefly discussed; in particular, that diffeomorphism invariance is not derivable as a gauge symmetry from the Hamiltonian formulation of tetrad gravity in any dimension when tetrads and spin connections are used as independent variables.

The results of high-precision variational calculations are reported for a number of bound states in various Coulomb three-body systems, including helium and helium-muonic atoms, some adiabatic systems (H2+, D 2+ and DT+ ions) and muonic molecular ions ppμ, ddμ, ttμ and dtμ. The hyperfine splittings for the double electron-excited states in the helium-muonic 3He2+ μ−e− and 4He2+ μ−e− atoms have also been determined. The results of this study are significantly more accurate than results known from earlier calculations for all considered systems and states. The approach can be used to determine the bound-state spectra in various three-body systems to arbitrary high accuracy. We also discuss a number of complications which are usually detected in high-precision bound-state calculations of few-body systems.

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