Vol. 32, No. 3, 1970

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Domain-perturbed problems for ordinary linear differential operators

John Froese

Vol. 32 (1970), No. 3, 651–662

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+). 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+. Further, asymptotic estimates are obtained for μabi λ as a,b ω+ The other results are refinements which lead to asymptotic estimates for the eigenfunctions and variational formulae for the eigenvalues.

Mathematical Subject Classification
Primary: 34.30
Secondary: 47.00
Received: 23 January 1968
Published: 1 March 1970
John Froese