The behavior of oscillatory integrals with degenerate stationary points

Abstract

(see Theorem 1.1 in §1). When the phase function {x) has only non-degenerate stationary points (in supp[/o]) or is analytic, I{a) is expanded asymptotically as |a|->oo (cf. Hormander [1], Varchenko [8], Kaneko [4], etc.),and then we can obtain the above estimate through that expansion. But it seems difficult to do so when all derivatives of $(x) vanish at some points. We take this case into consideration. In our methods we do not employ the asymptotic expansion. We can apply the above estimate to a scattering inverse problem studied by Ikawa [3]. Consider the scattering by obstacles formulated by Lax and Phillips[5]. Then the scattering matrix S(z) is meromorphic in the whole complex plane and analytic on the lower half plane {z: Imz^O} (cf. Chapter V of [5]). Ikawa [2], [3] examine the distribution of the poles of S(z) in the case where the obstacles consist of two convex bodies. In \2~] it is proved that if