Abstract
A spectral theory is deduced for differential eigenvalue problems related to a formally selfadjoint differential equation Su=λ Tu, where u is complex-valued and λ is the eigenvalue parameter. The equation or rather the differential operators S and T are considered on an arbitrary open interval of the real axis. The lower order operator T is assumed to have a positive definite Dirichlet integral which serves as scalar product in spectral theorems determined by symmetric boundary conditions. The theory is given in terms of ordered pairs u/\(\dot u\)of functions. Thus symmetric boundary conditions are certain subrelations of {u/\(\dot u\): Su=T\(\dot u\)}. If for instance T is the identity operator the boundary conditions are equally well described as conditions on u only. As far as the spectral theorem is concerned the method of the paper is easily transferred to the case when S instead of T has a positive definite Dirichlet integral. In the here considered case with T positive a kernel representation of the resolvent is deduced and used to prove the regularity of the elements of eigenspaces belonging to finite intervals of the spectral axis. The theory was worked out independently of the investigations by F.W. Schäfke, A. Schneider and H.-D. Niesser of systems of first order equations to which it seems related in different respects.
Keywords
- Continuous Derivative
- Spectral Theorem
- Finite Dimensional Subspace
- Linear Hull
- Symmetric Boundary Condition
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Bibliography
Bennewitz, Chr. and Pleijel, Åke. Selfadjoint extension of ordinary differential operators. To appear in the Proceedings of the Colloquium on Mathematical Analysis at Jyväskylä, Finland, August 17–21, 1970. (Mimeographed preprint available from the Department of Mathematics at Uppsala University.)
Niessen, Heinz-Dieter. Singuläre S-hermitesche Rand-Eigenwertprobleme. Manuscripta Mathematica, 3, 35–68, 1970. (Contains references to earlier papers by F.W. Schäfke and A. Schneider.)
Pleijel, Åke. Generalization of the spectral theory of ordinary formally self-adjoint differential operators. To appear in Journal of the Indian Mathematical Society, 34, 1972. (Mimeographed preprint available from the Department of Mathematics at Uppsala University.)
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© 1972 Springer-Verlag
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Pleijel, Å. (1972). Green's functions for pairs of formally selfadjoint ordinary differential operators. In: Everitt, W.N., Sleeman, B.D. (eds) Conference on the Theory of Ordinary and Partial Differential Equations. Lecture Notes in Mathematics, vol 280. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066924
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DOI: https://doi.org/10.1007/BFb0066924
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