Singularly Perturbed Eigenvalue Problems

This paper is concerned with eigenvalue problems of singularly perturbed linear ordinary differential equations. A common way to treat such problems is to derive an approximating eigenvalue problem by the use of matched asymptotic expansions. It is shown that under appropriate assumptions a domain in the complex plane can be identified, in which the eigenvalues of the approximating problem are isolated and that the eigenvalues and invariant subspaces of the singularly perturbed problem converge to these eigenvalues and corresponding invariant subspaces of the approximating eigenvalue problem as the perturbation parameter tends to zero. As an application a local stability analysis of the time-dependent semiconductor device equations via an eigenvalue problem is performed, and approximations of the eigenvalues in the case of a symmetric diode in the equilibrium state are computed.