ABSTRACT
We shall be concerned with the behaviour of eigenvalues of a boundary value problem in ordinary differential equation when a perturbation of form ε times a function of the independent variable x is introduced into one of the coefficients. As ε increases from 0, the perturbed eigenvalue λ(ε) describes a locus in the complex plane. In the case under consideration, the unperturbed problem is self-adjoint, with the result that the unperturbed eigenvalues lie on the real-axis. The effect of the perturbation is to destroy this self-adjointness, thus permitting the eigenvalues to move off the real-axis. There are several applications where this imaginary part of λ, although perhaps small compared with the perturbation in the real part, is of physical importance. Typically we might have Im λ(ε) = O(e −1/ε ) while λ ( ε ) = λ 0 + ε λ 1 + ε 2 λ 2 + ⋯ , λ k ∈ R , as ε → 0 + https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072584/300cbf75-7783-4a6f-9796-7cd4d736d320/content/eq3727.tif"/>
