Electronic ISBN:  9781470405175 
Product Code:  MEMO/195/911.E 
List Price:  $77.00 
MAA Member Price:  $69.30 
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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 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

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


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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.

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