Memoirs of the American Mathematical Society 2008; 177 pp; softcover Volume: 195 ISBN10: 0821841718 ISBN13: 9780821841716 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 twopoint 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
