Physics – Mathematical Physics
Scientific paper
2007-10-31
J. Math. Phys., vol. 49, 053523 (2008)
Physics
Mathematical Physics
27 pages, RevTeX, accepted for publication in J. Math. Phys
Scientific paper
10.1063/1.2919888
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 Fulling's 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 author's program in Mathematica, published in arXiv:0710.5406.
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