Eigenvalue curves and boundary conditions

Abstract

Nonlinear eigenvalue problems are considered which are defined by linear homogeneous systems of ordinary differential equations, subject to linear homogeneous boundary conditions, both depending analytically on a complex eigenvalue parameter ω, and on an additional small positive parameter ε such that all solutions have variation O(ε−1). There is, in general, a family of eigenvalues that, in the limit ε→0, become densely spaced and form a definite curve in the complex ω plane (exceptions arise only for some special boundary conditions). This eigenvalue curve is constructed for arbitrary nonsingular systems without turning points. It depends on no details of the boundary conditions other than their type (i.e., the numbers of boundary conditions involving only the left, only the right, or both end points of the interval). In the special cases of either only two-point boundary conditions (such as periodicity) or only one-point boundary conditions with equally many conditions at each end point, and of differential equations involving only even derivatives whose coefficients are constant to leading order, the curve is simply given by the roots of the local dispersion relation. The distance of the eigenvalues from the curve is O(ε) in general, but O(ε1/l log ε−1), with some positive integer l, in special cases.