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Operators on L2(I) ⊕ Cr

Published online by Cambridge University Press:  20 November 2018

R. R. D. Kemp
R. R. D. Kemp*
Affiliation:
Queen's University, Kingston, Ontario
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Several authors have considered eigenvalue problems for differential equations where the eigenvalue parameter also appears in the boundary conditions. Such problems do not appear to arise from any spectral problem associated with a linear operator on a Hilbert space. However, it is possible to reset such problems in this context. This has been done for certain second order cases by Walter [4] using a special measure on the interval in question, and by Fulton [1, 2] using the type of space indicated in the title of this article.

It is our purpose here to consider a general class of operators on L2(I) ⊕ Cr, which are based on a differential expression τ of order n on I. We shall first investigate adjoints, boundary conditions, and self-adjointness for such operators. We shall then show that all eigenvalue problems of the form τy = λy, with boundary conditions which involve λ in a linear fashion, can be reset in the context of such operators.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 1 , 01 February 1987 , pp. 33 - 53
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Fulton, C., Two point boundary value problems with the eigenvalue parameter contained in the boundary conditions, Proc. R.S.E. 77A, (1977), 293–308.Google Scholar
2. Fulton, C., Singular eigenvalue problems with the eigenvalue parameter contained in the boundary conditions, Proc. R.S.E. A87, (1980/81), 1–34.Google Scholar
3. Kemp, R. R. D. and Lee, S. J., Finite dimensional perturbations of differential expressions, Can. J. Math. 23 (1976), 1082–1104.Google Scholar
4. Walter, J., Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Math. Z. 133(1973), 301–312.Google Scholar