Asymptotic Variational Formulae for Eigenvalues

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Abstract
The eigenvalues of a second order self-adjoint elliptic differential operator on Riemannian n-space R will be considered. Our purpose is to obtain asymptotic variational formulae for the eigenvalues under the topological deformations of (i) removing an ɛ -cell (and adjoining an additional boundary condition on the boundary component thereby introduced); and (ii) attaching an ɛ -handle, valid on a half-open interval 0 < ɛ ≤ ɛo. In particular the formulae will exhibit the non-analytic nature of the variation. Similar variational problems for singular ordinary differential operators have been considered by the writer in [3] and [4].

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