Eigenvalues and completeness for regular and simply irregular two-point differential operators
著者
書誌事項
Eigenvalues and completeness for regular and simply irregular two-point differential operators
（Memoirs of the American Mathematical Society, no. 911）
American Mathematical Society, 2008
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注記
Includes bibliographical references (p. 171-173) and index
内容説明・目次
内容説明
In this monograph the author develops the spectral theory for an $n$th order two-point differential operator $L$ in the Hilbert space $L2[0,1]$, 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 $(\rhon I - \ell)u = 0$, 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 $(\rhon I - \ell)u = 0$, 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 $L2[0,1]$. He also gives examples of degenerate irregular differential operators illustrating some of the unusual features of this class.
目次
- Introduction
- Birkhoff approximate solutions
- The approximate characteristic determinant: Classification
- Asymptotic expansion of solutions
- The characteristic determinant
- The Green's function
- The eigenvalues for $n$ even}
- The eigenvalues for $n$ odd
- Completeness of the generalized eigenfunctions
- The case $L=T$, degenerate irregular examples
- Unsolved problems
- Appendix
- Bibliography
- Index
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