Lack of Critical Phase Points and Exponentially Faint Illumination Franco Cardin (cardin@math.unipd.it) and Alberto Lovison (lovison@math.unipd.it) Dipartimento di Matematica Pura ed Applicata Via Belzoni, 7 – 35131 Padova (Italy) Abstract. The Stationary Phase Principle (S.P.P.) states that in the computation of oscillatory integrals, the contributions of non stationary points of the phase are smaller than any power n of 1/k, for k →∞. Unfortunately, S.P.P. says nothing about the possible growth in the constants in the estimates with respect to the powers n. A quantitative estimate of oscillatory integrals with amplitude and phase in the Gevrey classes of functions shows that these contributions are asymptotically negligible, like exp(-ak b ), a,b> 0. An example in Optics is given. Keywords: Stationary Phase, Oscillatory Integrals, Wave Optics, Symplectic Ge- ometry, Lagrangian Submanifolds 1. Introduction An oscillatory integral is an integral of the form I (k) := Z u∈Ω a(u)e -ikϕ(u) du, Ω ⊆ R d , (1) where a and ϕ are C ∞ real functions, called respectively amplitude and phase, and k is a (large) parameter. They are typically employed to represent solutions for linear PDE’s depending on a real parameter, e.g. the Schr¨ odinger equation or the Helmholtz equation. A well known feature is the tight dependence on the values of a near the critical points of ϕ. More precisely, if in the domain of integration Ω there are no degenerate critical points for ϕ, the Stationary Phase Principle 1 holds, i.e. I (k) ’  2π k ¶ d 2 X u 0 :∇ϕ(u 0 )=0 a(u 0 ) exp {ikϕ(u 0 )} e i π 4 sgn(∇ 2 ϕ(u 0 )) p det ∇ 2 ϕ(u 0 ) . (2) In particular, a very standard argument, states that the contribu- tions to I coming from a compact subset K ⊂ Ω where there are no 1 This short-wave approximation is usually referred in physics as the WKB method, and it seems that it was first worked out by F. Carlini [4] –we learned it in [1]–, and later used by Kelvin, Stokes and many others in the 19th century. c  2004 Kluwer Academic Publishers. Printed in the Netherlands. Stime.tex; 1/09/2004; 13:19; p.1