Abstract
We study nonselfadjoint spectral problems for ordinary differential equationsN(y)−λP(y)=0 with λ-linear boundary conditions where the orderp of the differential operatorP is less than the ordern ofN. The present paper addresses the question of the completeness of the eigenfunctions and associated functions in the Sobolev spacesW k2 (0,1) fork=0,1,...,n. To this end we associate a pencil\(\mathbb{K}\)−λℍ of operators acting fromL 2(0,1) to the larger spaceL 2x(0,1)ℂn with the given problem. We establish completeness results for normal problems in certain finite codimensional subspaces ofW k2 (0,1) which are characterized by means of Jordan chains in 0 of the adjoint of the compact operator\(\mathbb{A}\)=ℍ\(\mathbb{K}\) −1.
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Dedicated to Professor Heinz Langer on the occasion of his 60th birthday
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Tretter, C. Nonselfadjoint spectral problems for linear pencilsN-λP of ordinary differential operators with λ-linear boundary conditions: Completeness results. Integr equ oper theory 26, 222–248 (1996). https://doi.org/10.1007/BF01191859
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DOI: https://doi.org/10.1007/BF01191859