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1.—On Square-integrable Solutions of Symmetric Systems of Differential Equations of Arbitrary Order.*

Published online by Cambridge University Press:  14 February 2012

V. I. Kogan  and
F. S. Rofe-Beketov
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Synopsis

The following results are obtained for symmetric differential expressions of arbitrary order r ≧ 1 with matrix coefficients on the half-line, with a non-negative (and possibly identically degenerate) weight matrix W(t), and with a spectral parameter λ: upper and lower bounds for the deficiency indices N(λ) are found; it is proved that N(λ) is independent of λ for Im λ< 0and for Im λ>0; under very general conditions it is proved that the maximum possible values of the deficiency indices in the half-planesIm λ≷0 can only be attained simultaneously; sufficient conditions for first-order expressions to be quasi-regular are derived; and it isshown that a symmetric system of any order reduces to a canonical, first-order system.

Examples are constructed, and the case of the whole line is also touched on.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 74 , 1976 , pp. 5 - 40
Copyright
Copyright © Royal Society of Edinburgh 1976

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