[en] Let E(R) be a potential-energy function for a neutral or ionic diatom in the Born-Oppenheimer approximation. Then the approximation of E(R) for 0 < R < infinity starting from finite and typically small sets of given information is considered. The approach is based on the fact that the scaled potential curves F(R) = R²E(R) derive from an eigenvalue problem which depends linearly on R. The nature of the curves F(R) is examined in detail. The results include the discovery of various approximants, some of which display rigorous bounding properties and others, closely related, whose behavior with respect to the approximated function appears to be predictable. 36 references

No abstract available

[en] The application of Barta's method (C.R. Acad. Sci. Paris. 204:472 (1937)) to Schroedinger's equation is reviewed, the result of Duffin (Phys. Rev.; 71:827 (1947)) being extended in several directions. (author)

No abstract available

[en] Complementary variational functionals are derived which impose upper and lower bounds on transition probabilities. These functionals are used to yield bounds in terms of sets of sum rules, and illustrative calculations are presented for hydrogen, helium and krypton atoms. 14 refs. (U.S.)

[en] A two parameter semigroup of linear operators in a Hilbert space is shown to commute in the Tjon-Wu model Boltzmann equation (TW equation). Under appropriate conditions on the parameters the operator is of Hilbert-Schmidt type. In this case, it generates new solutions of the TW equation from known ones; and different existence domains can be derived from those already established. Analogous transformations may be obtained for other Boltzmann equations which are related to the TW equation via integral transforms. (author)

[en] Two fully nonlinear Boltzmann equations for infinite spatially uniform gases consisting of a single species of particle are considered. The models are the one due to Bobylev, Krook and Wu and the one due to Tjon and Wu. These are shown to be related by the Abel transform and an existence uniqueness theorem is presented. Then a procedure for the numerical calculations of solutions - based upon the method of Barnsley and Cornille - is given and a device for speeding up its convergence is described. Finally, several numerical examples are discussed. Some enhancement is shown to occur, but not enough to be physically important. (author)