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A phase of the differential equation \(y''=Q(t)y\) with a complex coefficient Q of the real variable. (English) Zbl 0645.34007

Let Q be a continuous complex-valued function on \(j=(a,b)\) \((-D\leq a<b\leq \infty)\), Im Q(t)\(\not\equiv 0\). Then there exist independent solutions \(y_ 1,y_ 2\) of (Q): \(y''=Q(t)y\) such that \(y^ 2_ 1(t)+y^ 2_ 2(t)\neq 0\) for \(t\in j\). We say that a \(C^ 3\) complex-valued function \(\alpha\) is a (first) phase of (Q) if there exist independent solutions u, v of (Q), \(u^ 2(t)+v^ 2(t)\neq 0\) on j, such that \(\alpha '(t)=- w/(u^ 2(t)+v^ 2(t)),\) \(t\in j\), where \(w=uv'-u'v\). In the real case the phase of the second-order differential equation was introduced by O. Bor\D{u}vka [Linear Differential Transformations of the Second Order (1971; Zbl 0222.34002)]. If \(\alpha\) is a phase of (Q) then \(b^{i\alpha (t)}/\sqrt{\alpha'(t)}\), \(e^{-i\alpha (t)}/\sqrt{\alpha'(t)}\) are independent solutions of (Q). Phases of (Q) are used for the investigation of the decomposition of zeros of solutions to the differential equation (Q).
Reviewer: S.Staněk

MSC:

34A30 Linear ordinary differential equations and systems
34M99 Ordinary differential equations in the complex domain

Citations:

Zbl 0222.34002
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References:

[1] Borůvka O.: Linear Differential Transformations of the Second Order. The English Univ. Press, London, 1971. · Zbl 0218.34005
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