Geodesic
Asymptotic Variational Formulae for Eigenvalues
Canadian mathematical bulletin, Tome 6 (1963) no. 1, pp. 15-25

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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]. 
Swanson, C.A. Asymptotic Variational Formulae for Eigenvalues. Canadian mathematical bulletin, Tome 6 (1963) no. 1, pp. 15-25. doi: 10.4153/CMB-1963-004-9
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     author = {Swanson, C.A.},
     title = {Asymptotic {Variational} {Formulae} for {Eigenvalues}},
     journal = {Canadian mathematical bulletin},
     pages = {15--25},
     year = {1963},
     volume = {6},
     number = {1},
     doi = {10.4153/CMB-1963-004-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1963-004-9/}
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