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

John Locker Colorado State University, Fort Collins, CO
Available Formats:
Electronic ISBN: 978-1-4704-0517-5
Product Code: MEMO/195/911.E
List Price: $77.00 MAA Member Price:$69.30
AMS Member Price: $46.20 Click above image for expanded view Eigenvalues and Completeness for Regular and Simply Irregular Two-Point Differential Operators John Locker Colorado State University, Fort Collins, CO Available Formats:  Electronic ISBN: 978-1-4704-0517-5 Product Code: MEMO/195/911.E  List Price:$77.00 MAA Member Price: $69.30 AMS Member Price:$46.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 1952008; 177 pp
MSC: Primary 34; Secondary 47;

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.

• Chapters
• 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
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Volume: 1952008; 177 pp
MSC: Primary 34; Secondary 47;

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.

• Chapters
• 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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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