Phase Integral Approximation for coupled ODEs of the Schroedinger type

Four generalizations of the Phase Integral Approximation (PIA) to sets of N ordinary differential equations of the Schroedinger type: u_j(x) + Sum{k = 1 to N} R_{jk}(x) u_k(x) = 0, j = 1 to N, are described. The recurrence relations for higher order corrections are given in the form valid in arbitrary order and for the matrix R_{jk} either hermitian or non-hermitian. For hermitian and negative definite R matrices, the Wronskian conserving PIA theory is formulated which generalizes Fullings current conserving theory pertinent to positive definite R matrices. The idea of a modification of the PIA, well known for one equation: u(x) + R(x) u(x) = 0, is generalized to sets. A simplification of Wronskian or current conserving theories is proposed which in each order eliminates one integration from the formulas for higher order corrections. If the PIA is generated by a non-degenerate eigenvalue of the R matrix, the eliminated integration is the only one present. In that case, the simplified theory becomes fully algorithmic and is generalized to non-hermitian R matrices. General theory is illustrated by a few examples generated automatically by using authors program in Mathematica, published in arXiv:0710.5406.