# Eigenvalues and Completeness for Regular and Simply Irregular Two-Point Differential Operators

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*John Locker*

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

# Table of Contents

## Eigenvalues and Completeness for Regular and Simply Irregular Two-Point Differential Operators

- Contents v6 free
- Chapter 1. Introduction 110 free
- Chapter 2. Birkhoff Approximate Solutions 1322
- Chapter 3. The Approximate Characteristic Determinant: Classification 1928
- Chapter 4. Asymptotic Expansion of Solutions 3948
- Chapter 5. The Characteristic Determinant 5766
- Chapter 6. The Green's Function 7584
- Chapter 7. The Eigenvalues for n Even 99108
- Chapter 8. The Eigenvalues for n Odd 119128
- Chapter 9. Completeness of the Generalized Eigenfunctions 139148
- Chapter 10. The Case L = T, Degenerate Irregular Examples 147156
- Chapter 11. Unsolved Problems 165174
- Chapter 12. Appendix 169178
- Bibliography 171180
- Index 175184 free