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Domain-perturbed problems for ordinary linear differential operators. Froese, John

Abstract

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_ < a < 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 [ Formula omitted ] as a,b ⇢ ω_,ω+. Further, asymptotic estimates are obtained for [ Formula omitted ] as a,b ⇢ ω_,ω+. The other results are refinements which lead to asymptotic estimates for the eigenfunctions and variational formulae for the eigenvalues. The conditions on the limiting behaviour of the boundary operators depend strongly on the nature of the singularities ω_,ω+. If for some numberℓ⃘, ℓ⃘not an eigenvalue, linearly independent solutions of Lx = ℓ⃘x exist which are asymptotically ordered at ω_, then ω_ is palled a class 1 singularity. In the case that both ω_,ω+ are class 1 singularities, very general boundary operators permit the convergence of [ Formula omitted ], Class 2 singularities are defined as follows: If all solutions of Lx = ℓ⃘x are square-integrable on (ω_, c] for any c satisfying ω_ < c < ω+, then ω_ is called a class 2 singularity. An asymptotic ordering of the solutions is not assumed in this case. Since the behaviour of the solutions of Lx = ℓ⃘x is essentially arbitrary when both ω_,ω+ are class 2 singularities, the generality of the boundary operators has to be sacrificed to ensure that [ Formula omitted ]. Certain one end perturbation problems and examples also are considered.

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