Geodesic
Splitting problem for WKB asymptotics in a~nonresonant case and the reduction method for linear systems
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Order and chaos in dynamical systems, Tome 297 (2017), pp. 292-312

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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). 
@article{TM_2017_297_a15,
     author = {S. A. Stepin},
     title = {Splitting problem for {WKB} asymptotics in a~nonresonant case and the reduction method for linear systems},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {292--312},
     publisher = {mathdoc},
     volume = {297},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2017_297_a15/}
}
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S. A. Stepin. Splitting problem for WKB asymptotics in a~nonresonant case and the reduction method for linear systems. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Order and chaos in dynamical systems, Tome 297 (2017), pp. 292-312. http://geodesic.mathdoc.fr/item/TM_2017_297_a15/