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The quasilinearization method (QLM) of solving nonlinear differential equations is applied to the quantum mechanics by casting the Schrödinger equation in the nonlinear Riccati form. The method, whose mathematical basis in physics was discussed recently by one of the present authors (VBM), approaches the solution of a nonlinear differential equation by approximating the nonlinear terms by a sequence of the linear ones, and is not based on the existence of some kind of a small parameter. It is shown that the quasilinearization method gives excellent results when applied to computation of ground and excited bound state energies and wave functions for a variety of the potentials in quantum mechanics most of which are not treatable with the help of the perturbation theory or the 1/N expansion scheme. The convergence of the QLM expansion of both energies and wave functions for all states is very fast and already the first few iterations yield extremely precise results. The precison of the wave function is typically only one digit inferior to that of the energy. In addition it is verified that the QLM approximations, unlike the asymptotic series in the perturbation theory and the 1/N expansions are not divergent at higher orders.

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Matrix elements of two-body potentials and correlation functions between threeand four-body hyperspherical states, including their velocity-dependent parts, are calculated analytically for any value of the total orbital angular momentum. The resulting formulas contain explicitly written functions of the radial variable, and the Raynal-Revai coefficients. The latter are expressible through finite sums of 3-j and 9-j symbols. The formulas allow precise and fast evaluation of matrix elements of the effective potential in the correlation-function hyperspherical-harmonic method for atomic, molecular, and nuclear threeand four-body problems. The generalization to any number of particles is straightforward.