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

by Locker, John

• ISBN13: 9780821841716
• ISBN10: 0821841718
• Format: Paperback
• Copyright: 2008-08-08
• Publisher: Amer Mathematical Society
• More Book Details

Summary

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 threepossible 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.

Table of Contents

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

Table of Contents provided by Publisher. All Rights Reserved.