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Resurgent equations and semiclassical asymptotics. (English. Russian original) Zbl 0929.34047
Differ. Equations 33, No. 10, 1372-1379 (1997); translation from Differ. Uravn. 33, No. 10, 1366-1373 (1997).
The paper is devoted to constructing exact asymptotics for the ordinary differential equation \[ \sum^n_{j=0} A_j(x)\left(-ih {d\over dx} \right) u(x,h)=0 \] with the polynomial coefficients \(A_j(x)\). The authors seek asymptotic solutions to this equation in the form of power series in \(h\) \[ u(x, h) \cong e^{(i/h) S_1(x)}\sum^\infty_{k=0} (-ih)^k a_k^{(1)}(x)+ e^{ (i/h)S_2(x)} \sum^\infty_{k=0} (-ih)^ka_k^{(2)} (x)+\cdots \] for complex values of \(x\). The determination of these series includes two stages:
(1) by using the theory of the Hamilton-Jacobi equation they construct formal WKB-elements which are formal solutions to the above equation up to an arbitrary power of \(h\);
(2) the exact asymptotic approximations to the considered equation are found on the basis of resummation of such formal solutions.
This procedure is illustrated by examples of the Airy equation and the Weber one.
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34M37 Resurgence phenomena (MSC2000)
47E05 General theory of ordinary differential operators
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.