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Eigenvalues and Completeness for Regular and Simply Irregular Two-Point Differential Operators
John Locker, Colorado State University, Fort Collins, CO
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Memoirs of the American Mathematical Society
2008; 177 pp; softcover
Volume: 195
ISBN-10: 0-8218-4171-8
ISBN-13: 978-0-8218-4171-6
List Price: US$73 Individual Members: US$43.80
Institutional Members: US\$58.40
Order Code: MEMO/195/911

In this monograph the author develops the spectral theory for an $$n$$th order two-point differential operator $$L$$ in the Hilbert space $$L^2[0,1]$$, where $$L$$ is determined by an $$n$$th order formal differential operator $$\ell$$ having variable coefficients and by $$n$$ linearly independent boundary values $$B_1, \ldots, B_n$$. Using the Birkhoff approximate solutions of the differential equation $$(\rho^n I - \ell)u = 0$$, the differential operator $$L$$ is classified as belonging to one of three possible classes: regular, simply irregular, or degenerate irregular. For the regular and simply irregular classes, the author develops asymptotic expansions of solutions of the differential equation $$(\rho^n I - \ell)u = 0$$, constructs the characteristic determinant and Green's function, characterizes the eigenvalues and the corresponding algebraic multiplicities and ascents, and shows that the generalized eigenfunctions of $$L$$ are complete in $$L^2[0,1]$$. He also gives examples of degenerate irregular differential operators illustrating some of the unusual features of this class.

• Introduction
• Birkhoff approximate solutions
• The approximate characteristic determinant: Classification
• Asymptotic expansion of solutions
• The characteristic determinant
• The Green's function
• The eigenvalues for $$n$$ even
• The eigenvalues for $$n$$ odd
• Completeness of the generalized eigenfunctions
• The case $$L=T$$, degenerate irregular examples
• Unsolved problems
• Appendix
• Bibliography
• Index