International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 13, Pages 647-678
doi:10.1155/S0161171204303522
Geometric aspects of high-order eigenvalue problems I. Structures on spaces of boundary conditions
1Department of Mathematics, Yangzhou University, Jiangsu, Yangzhou 225002, China
2Department of Mathematical Sciences, Northern Illinois University, DeKalb 60115, IL, USA
Received 11 March 2002; Revised 3 November 2003
Copyright © 2004 Xifang Cao and Hongyou Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We consider some geometric aspects of regular eigenvalue problems of an arbitrary order. First, we clarify a natural geometric structure on the space of boundary conditions. This structure is the base for studying the dependence of eigenvalues on the boundary condition involved, and reveals new properties of these eigenvalues. Then, we solve the selfadjointness condition explicitly and obtain a manifold structure on the space of selfadjoint boundary conditions and several other consequences. Moreover, we give complete characterizations of several subsets of boundary conditions such as the set of all complex boundary conditions having a given complex number as an eigenvalue, and describe some of them topologically. The shapes of some of these subsets are shown to be independent of the quasidifferential equation in question.