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Spectral Theory of Non-Self-Adjoint Two-Point Differential Operators
John Locker, Colorado State University, Fort Collins, CO
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Mathematical Surveys and Monographs
2000; 252 pp; hardcover
Volume: 73
ISBN-10: 0-8218-2049-4
ISBN-13: 978-0-8218-2049-0
List Price: US$75 Member Price: US$60
Order Code: SURV/73

This monograph develops the spectral theory of an $$n$$th order non-self-adjoint two-point differential operator $$L$$ in the Hilbert space $$L^2[0,1]$$. The mathematical foundation is laid in the first part, where the spectral theory is developed for closed linear operators and Fredholm operators. An important completeness theorem is established for the Hilbert-Schmidt discrete operators. The operational calculus plays a major role in this general theory.

In the second part, the spectral theory of the differential operator $$L$$ is developed by expressing $$L$$ in the form $$L = T + S$$, where $$T$$ is the principal part determined by the $$n$$th order derivative and $$S$$ is the part determined by the lower-order derivatives. The spectral theory of $$T$$ is developed first using operator theory, and then the spectral theory of $$L$$ is developed by treating $$L$$ as a perturbation of $$T$$. Regular and irregular boundary values are allowed for $$T$$, and regular boundary values are considered for $$L$$. Special features of the spectral theory for $$L$$ and $$T$$ include the following: calculation of the eigenvalues, algebraic multiplicities and ascents; calculation of the associated family of projections which project onto the generalized eigenspaces; completeness of the generalized eigenfunctions; uniform bounds on the family of all finite sums of the associated projections; and expansions of functions in series of generalized eigenfunctions of $$L$$ and $$T$$.

Graduate students and research mathematicians interested in ordinary differential equations.

Reviews

"An up-to-date account of the spectral theory of non-self-adjoint ordinary differential equations on a compact interval of the real line ... This book is well written and is accessible to all who have a rudimentary knowledge of functional analysis. It is well suited both to graduate students working in two-point boundary value problems and to other scientists seeking further information concerning them."

-- Bulletin of the LMS

"Detailed proofs are given; this, together with the introductory material of the first two chapters, should make the book accessible to a good graduate student with some background in functional analysis and differential equations. The monograph will probably appeal mainly to specialists."

-- Mathematical Reviews