Adiabatic expansions of solutions of coupled second‐order linear differential equations. II

摘要: The ’’phase‐integral’’ approach to higher‐order WKB approximations, associated with the names of Froman and Chakraborty, is generalized systems equations, written in vector notation as h″ (t)+u2 M(t)h(t) =0, where M positive definite u→∞. Expansions are constructed form h∼p−1/4e exp (−iuFt p−1/2 dt′), p e power series u−1 (asymptotically) a unit vector. Expanding out terms exponential yields approximations studied first paper this series, which less uniform t. (A search for still greater uniformity leads nonlinear differential equations leading term e, can be explicitly solved only special cases, notably that all eigenvalues distinct.) expansions proved valid asymptotic on finite intervals strictly do not cross (i.e., multiplicity does depend t).