Real-parameter square-integrable solutions and the spectrum of differential operators |
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Authors: | Xiaoling Hao |
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Institution: | a Math. Dept., Inner Mongolia University, Hohhot 010021, China b Math. Dept., Northern Illinois University, DeKalb, IL 60115, USA |
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Abstract: | We continue to investigate the connection between the spectrum of self-adjoint ordinary differential operators with arbitrary deficiency index d and the number of linearly independent square-integrable solutions for real values of the spectral parameter λ. We show that if, for all λ in an open interval I, there are d linearly independent square-integrable solutions, then there is no continuous spectrum in I. This for any self-adjoint realization with boundary conditions which may be separated, coupled, or mixed. The proof is based on a new characterization of self-adjoint domains and on limit-point (LP) and limit-circle (LC) solutions established in an earlier paper. |
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Keywords: | Differential operators Continuous spectrum Deficiency index Singular boundary conditions |
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