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On subordinacy and spectral multiplicity for a class of singular differential operators

Published online by Cambridge University Press:  14 November 2011

D. J. Gilbert
D. J. Gilbert
Affiliation:
School of Science and Mathematics, Sheffield Hallam University, Sheffield SI 1WB, U.K.
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Abstract

The spectral multiplicity of self-adjoint operators H associated with singular differential expressions of the form

is investigated. Based on earlier work of I. S. Kac and recent results on subordinacy, complete sets of necessary and sufficient conditions for the spectral multiplicity to be one or two are established in terms of: (i) the boundary behaviour of Titchmarsh–Weyl m-functions, and (ii) the asymptotic properties of solutions of Lu = λu, λ∈ℝ, at the endpoints a and b. In particular, it is shown that H has multiplicity two if and only if L is in the limit point case at both a and b and the set of all λ for which no solution of Lu = λu is subordinate at either a or b has positive Lebesgue measure. The results are completely general, subject only to minimal restrictions on the coefficients p(r), q(r)and w(r), and the assumption of separated boundary conditions when L is in the limit circle case at both endpoints.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 3 , 1998 , pp. 549 - 584
Copyright
Copyright © Royal Society of Edinburgh 1998

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