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Spiral motion for components of a Dirac system of limit circle type

Published online by Cambridge University Press:  14 November 2011

S. G. Halvorsen  and
J. K. Shaw
S. G. Halvorsen
Affiliation:
Department of Mathematics, University of Trondheim, 7000 Trondheim, Norway
J. K. Shaw
Affiliation:
Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061–4097, U.S.A.
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Synopsis

The components of a two (complex) dimensional Dirac system are studied as trajectories in the complex plane. The system, defined on an interval [a, b) of regular points, is assumed to be of limit circle type at t = b, and to be non-oscillatory there. By introducing moduli ρ1(t) = |y1(t)|, ρ2(t) = |y2(t)| and continuous complex arguments θ1(t) = arg y1(t), θ2(t) = arg y2(t) for the components, the principal result proved is that θ1(t) and θ2(t) are bounded as t → b. Examples show that monotonicity of the argument function θ1(t), which is a feature of the corresponding problem for Sturm–Liouville equations, fails for Dirac systems.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 97 , 1984 , pp. 97 - 104
Copyright
Copyright © Royal Society of Edinburgh 1984

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