A mathematical treatment is presented of the known Berry, Wilczek-Zee, Aharonov-Anandan, and Pancharatnam topological phases, and simple illustrative examples of their quantum mechanics are presented. The continuity and connection is traced among the various phases, while filling the gap involved with the forgotten works of S. M. Rytov and V. V. Vladimirskii in polarization optics. A set of current experiments in polarization optics where the topological phases are measured is discussed in detail. Additional information on recently obtained results involving manifestations of geometrical phases in quantum mechanics and other fields of physics is contained in the Appendix and in the studies cited there.

For the three-body Coulomb problem a hyperspherical parametrisation of independent variables is given on a five-dimensional sphere S5 with a hyperradius RH, the first linear invariant of the inertia tensor. The hyperspherical adiabatic basis is defined as a complete set of eigenfunctions and eigenvalues of the Hamiltonian on the sphere S5 for every fixed value of the slow variable RH. The partial wave analysis in the total momentum J representation allows the authors to separate three Euler angles and to reduce the hyperspherical problem on S5 to a system of (J+1) two-dimensional problems. Classification is given of the hyperspherical adiabatic basis for small and large values of the hyperradius RH. The logarithmic Fock singularity at the point of triple collision (RH=0) is explicitly shown. The approach is assigned to computing the cross sections of mesic atomic processes in the muon catalysis problem.

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The binding energy for excitonic molecules, four-body systems with Coulomb interaction, is investigated using the adiabat,ic method usually applied to three-body Coulomb systems. The infinite system of coupled Schrodinger-type equations obtained, is approximated by a finite system of twenty-one coupled equations and solved. The investigation gives an increase in binding of these systems practically in the whole region 0 < CT = me/mh 5 1, compared to previous variational results.Die Bindungsenergie fur Exzitonenmolekule, Vierkorpersysteme mit Coulomb-Wechselwirkung, wird mit der adiabatischen Methode untersucht, die gewohnlich fur Drei-Korper-Coulombsysteme benutzt wird. Das erhaltene unendliche System gekoppelter Gleichungen vom Schriidingertyp wird durch ein endliches System von einundzwanzig gekoppelten Gleichungen genahert und gelost. Die Untersuchung ergibt ein Anwachsen der Bindnng dieses Systems praktisch im gesamten Bereich 0 < u = me/mh 5 1, verglichen zu friiheren Variationsergebnissen.

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