Abstract
The spectral multiplicity of self-adjoint operatorsHassociated with singular differential expressions of the formis 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–Weylm-functions, and (ii) the asymptotic properties of solutions ofLu= λu, λ∈ℝ, at the endpointsaandb. In particular, it is shown thatHhas multiplicity two if and only ifLis in the limit point case at bothaandband the set of all λ for which no solution ofLu= λuis subordinate at eitheraorbhas positive Lebesgue measure. The results are completely general, subject only to minimal restrictions on the coefficientsp(r), q(r)andw(r), and the assumption of separated boundary conditions whenLis in the limit circle case at both endpoints.
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