## Solutions to the operator equation $$i\varepsilon dy/dt=A(t)y$$ on intervals containing turning points.(English. Russian original)Zbl 0964.34039

Theor. Math. Phys. 122, No. 3, 298-311 (2000); translation from Teor. Mat. Fiz. 122, No. 3, 357-371 (2000).
The author studies the asymptotic behavior as $$\varepsilon \rightarrow 0$$ of the solution to the equation $i \varepsilon \frac{dy}{dt} =A(t)y ,$ where $$A(t)$$ is a linear closed operator defined on a dense subset of a Banach space $$X$$. The solutions to the above equation are constructed as formal asymptotic expansions as $$\varepsilon \rightarrow 0$$ on intervals containing parabolic or hyperbolic turning points. A recursive scheme for finding the succesive terms of these expansions are obtained.

### MSC:

 34D05 Asymptotic properties of solutions to ordinary differential equations 34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations 34G10 Linear differential equations in abstract spaces 34E05 Asymptotic expansions of solutions to ordinary differential equations
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### References:

 [1] V. Buslaev and A. Grigis, ”Turning points,” Preprint Math. 98-21. l’Université Paris-Nord, Paris (1998). · Zbl 0987.35013 [2] Yu. L. Daletskii and M. G. Krein,Stability of Solutions of Differential Equations in Banach Space [in Russian], Nauka, Moscow (1972); English transl. (Transl. Math. Monographs, Vol. 43), Am. Math. Soc., Providence, RI (1974). [3] S. G. Krein,Linear Differential Equations in Banach space [in Russian], Nauka, Moscow (1969); English transl., Am. Math. Soc., New York (1972). · Zbl 0188.41401 [4] V. S. Buldyrev and S. Yu. Slavyanov, ”Regularization of phase integrals near the barrier top [in Russian],” in:Problems of Mathematical Physics, Vol. 10,Spectral Theory. Wave Processes (M. Birman, et al., eds.), Izd. Leningradskogo Universiteta, Leningrad (1982), pp. 50–70. · Zbl 0513.70016
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