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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
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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.

Table of Contents

  • 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
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