Miscellaneous

Abstract

The beautiful and historical note of Louis [B78], is about one of the most important results on the behavior in the complex domain of the eigenfunctions of a self-adjoint non negative elliptic operator P on a real analytic compact manifold M. Let (f _k ) be an orthonormal basis of eigenfunctions of P with Pf _k = λ _k f _k . Under a convexity assumption on the principal symbol p(x, ξ) of P, the result asserts that there exists a natural family of strictly pseudo-convex tubular neighborhoods B _ɛ of M in its complexification X such that a function f = ∑ _k a _k f _k on M extends holomorphically in B _ɛ , with a L ² boundary value on ∂ B _ɛ , iff the series $\sum _{k}\lambda _{k}^{-(n-1)/2}e^{2\varepsilon \lambda _{k}}\vert a_{ k}\vert ^{2}$ is convergent. This result applies in particular to the Laplace operator in the Riemannian case. This work of Louis has been a source of inspiration for many authors, and is the starting point of the numerous applications of the properties of analytic continuation of eigenfunctions: nodal geometry, analytic wave front sets, tunneling estimates, random-waves…The proof of the above result suggested by Louis involves FIO’s in the complex domain.