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We establish the existence of analytic curves of eigenvalues for theLaplace-Neumann operator through an analytic variation of the metric of acompact Riemannian manifold $M$ with boundary by means of a new approach ratherthan Kato's method for unbounded operators. We obtain an expression for thederivative of the curve of eigenvalues, which is used as a device to prove thatthe eigenvalues of the Laplace-Neumann operator are generically simple in thespace $\mathcal{M}^k$ of all $C^k$ Riemannian metrics on $M$. This implies theexistence of a residual set of metrics in $\mathcal{M}^k$, which make thespectrum of the Laplace-Neumann operator simple. We also give a preciseinformation about the complementary of this residual set, as well as about thestructure of the set of the deformation of a Riemannian metric which preservesdouble eigenvalues.

On Eigenvalue Generic Properties of the Laplace-Neumann Operator