The variation of the eigenvalues
and eigenfunctions of an ordinary linear self-adjoint differential operator L is
considered under perturbations of the domain of L. The basic problem is defined as a
suitable singular eigenvalue problem for L on the open interval ω−< s < ω+ and is
assumed to have at least one real eigenvalue λ of multiplicity k. The perturbed
problem is a regular self-adjoint problem defined for L on a closed subinterval
[a,b] of (ω−,ω+). It is proved under suitable conditions on the boundary
operators of the perturbed problem that exactly k perturbed eigenvalues
μabi→ λ as a,b → ω−,ω+. Further, asymptotic estimates are obtained for
μabi− λ as a,b → ω−,ω + ⋅ The other results are refinements which lead to
asymptotic estimates for the eigenfunctions and variational formulae for the
eigenvalues.
