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Splitting problem for WKB asymptotics in a nonresonant case and the reduction method for linear systems

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Abstract

As applied to the problem of asymptotic integration of linear systems of ordinary differential equations, we propose a reduction of order method that allows one to effectively construct solutions indistinguishable in the growth/decrease rate at infinity. In the case of a third-order equation, we use the developed approach to answer Bellman’s problem on splitting WKB asymptotics of subdominant solutions that decrease at the same rate. For a family of Wigner–von Neumann type potentials, the method allows one to formulate a selection rule for nonresonance values of the parameters (for which the corresponding second-order equation has a Jost solution).

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991, Russia

    S. A. Stepin

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  1. S. A. Stepin
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Correspondence to S. A. Stepin.

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Original Russian Text © S.A. Stepin, 2017, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2017, Vol. 297, pp. 292–312.

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Stepin, S.A. Splitting problem for WKB asymptotics in a nonresonant case and the reduction method for linear systems. Proc. Steklov Inst. Math. 297, 264–284 (2017). https://doi.org/10.1134/S0081543817040162

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  • DOI: https://doi.org/10.1134/S0081543817040162

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