Locker, John, author2015-12-092015-12-092006http://hdl.handle.net/10217/170086August 29, 2006.In this monograph the author develops the spectral theory for an nth order two-point differential operator L in the Hilbert space L2[0,1], where L is determined by an nth order formal differential operator ℓ having variable coefficients and by n linearly independent boundary values B1,…,Bn. Using the Birkhoff approximate solutions of the differential equation (ρnI−ℓ)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 (ρnI−ℓ)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 L2[0,1]. He also gives examples of degenerate irregular differential operators illustrating some of the unusual features of this class.1. Introduction -- 2. Birkhoff approximate solutions -- 3. The approximate characteristic determinant: classification -- 4. Asymptotic expansion of solutions -- 5. The characteristic determinant -- 6. The Green's function -- 7. The eigenvalues for n even -- 8. The eigenvalues for n odd -- 9. Completeness of the generalized eigenfunctions -- 10. The case L = T, degenerate irregular examples -- 11. Unsolved problems -- 12. Appendix.born digitalbooksengEigenvaluesDifferential operatorsTextThis book is open access and distributed under the terms and conditions of the Creative Commons Attribution 4.0 International (CC BY 4.0).