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Register a new userPrograms in Mathematica relevant to Phase Integral Approximation for coupled ODEs of the Schrodinger type
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Authors
Andrzej A. Skorupski
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Three programs in Mathematica are presented, which produce expressions for the lowest order and the higher order corrections of the Phase Integral Approximation. First program is pertinent to one ordinary differential equation of the Schrodinger type. The remaining two refer to a set of two such equations.
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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.
The general construction of quasi-classically concentrated solutions to the Hartree-type equation, based on the complex WKB-Maslov method, is presented. The formal solutions of the Cauchy problem for this equation, asymptotic in small parameter h (hto0), are constructed with a power accuracy of O(h^{N/2}), where N is any natural number. In constructing the quasi-classically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for middle or centered moments) is essentially used. The nonlinear superposition principle has been formulated for the class of quasi-classically concentrated solutions of the Hartree-type equations. The results obtained are exemplified by the one-dimensional equation Hartree-type with a Gaussian potential.Comments: 6 pages, 4 figures, LaTeX Report no: Subj-class: Accelerator Physics
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