## 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

Zbl 0222.34002
Full Text:

### References:

 [1] Borůvka O.: Linear Differential Transformations of the Second Order. The English Univ. Press, London, 1971. · Zbl 0218.34005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.