A bundle view of boundary-value problems: generalizing the Gardner-Jones bundle

Holomorphic families of linear ordinary differential equations on a finite interval with prescribed parameter-dependent boundary conditions are considered from a geometrical viewpoint. The Gardner-Jones bundle, which was introduced for linearized reaction-diffusion equations, is generalized and applied to this abstract class of λ-dependent boundary-value problems, where λ is a complex eigenvalue parameter. The fundamental analytical object of such boundary-value problems (BVPs) is the characteristic determinant, and it is proved that any characteristic determinant on a Jordan curve can be characterized geometrically as the determinant of a transition function associated with the Gardner-Jones bundle. The topology of the bundle, represented by the Chern number, then yields precise information about the number of eigenvalues in a prescribed subset of the complex λ-plane. This result shows that the Gardner-Jones bundle is an intrinsic geometric property of such λ-dependent BVPs. The bundle framework is applied to examples from hydrodynamic stability theory and the linearized complex Ginzburg-Landau equation.