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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

An eigenfunction expansion for a nonselfadjoint, interior point boundary value problem


Author: Allan M. Krall
Journal: Trans. Amer. Math. Soc. 170 (1972), 137-147
MSC: Primary 34B25; Secondary 47E05
DOI: https://doi.org/10.1090/S0002-9947-1972-0311985-3
MathSciNet review: 0311985
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Abstract: Under discussion is the vector system $ Ly = y' + Py,\sum\nolimits_{j = 0}^\infty {{A_j}y({t_j}) = 0} $, where $ \sum\nolimits_{j = 0}^\infty {\vert\vert A\vert\vert < \infty } $. The eigenvalues for the system are known to be countable and approach $ \infty $ in the complex plane in a series of well-defined vertical steps. For each eigenvalue there exists an eigenmanifold, generated by the residue of the Green's function. Further, since the Green's function vanishes near $ \infty $ in the complex plane when the path toward $ \infty $ is horizontal, the Green's function can be expressed as a series of its residues. This in turn leads to two eigenfunction expansions, one for elements in the domain of the original system, another for elements in the domain of the adjoint system.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0311985-3
Keywords: Operator, differential operator, boundary value problem, differential system, nonselfadjoint, Sturm-Liouville problem, adjoint, eigenvalue, spectrum, spectral resolution
Article copyright: © Copyright 1972 American Mathematical Society