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Asymptotic form and infinite product representation of solution of a second order initial value problem with a complex parameter and a finite number of turning points. (English) Zbl 1302.34089

J. Contemp. Math. Anal., Armen. Acad. Sci. 46, No. 4, 212-226 (2011) and Izv. Nats. Akad. Nauk Armen., Mat. 46, No. 4, 57-76 (2011).
Summary: The paper studies the differential equation \[ y'' + (\rho ^2 \phi ^2 (x) - q(x))y = 0 \eqno{(*)} \] on the interval \(I = [0, 1]\), containing a finite number of zeros \(0 < x_{1} < x_{2} < \ldots < x_{m} < 1\) of \(\phi^{2}\), i.e. so-called turning points. Using asymptotic estimates from [W. Eberhard, G. Freiling and A. Schneider, “Connection formulae for second-order differential equations with a complex parameter and having an arbitrary number of turning points”, Math. Nachr. 165, No. 1, 205–229 (1994; doi:10.1002/mana.19941650114)] for appropriate fundamental systems of solutions of \((^*)\) as \(|\rho| \to \infty\), it is proved that, if there is an asymptotic solution of the initial value problem generated by \((^*)\) in the interval \([0, x_{1})\), then the asymptotic solutions in the remaining intervals can be obtained recursively. Furthermore, an infinite product representation of solutions of \((^*)\) is studied. The representations are useful in the study of inverse spectral problems for such equations.

MSC:

34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
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