| eBook 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 |
| eBook 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 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 195; 2008; 177 ppMSC: 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.
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Table of Contents
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Chapters
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1. Introduction
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2. Birkhoff approximate solutions
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3. The approximate characteristic determinant: classification
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4. Asymptotic expansion of solutions
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5. The characteristic determinant
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6. The Green’s function
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7. The eigenvalues for $n$ even
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8. The eigenvalues for $n$ odd
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9. Completeness of the generalized eigenfunctions
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10. The case $L$ = $T$, degenerate irregular examples
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11. Unsolved problems
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12. Appendix
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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.
-
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
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11. Unsolved problems
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12. Appendix
