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Generalized Fresnel Integrals Bull. Sci. math. 129 (2005) 1–23 www.elsevier.com/locate/bulsci Generalized Fresnel integrals S. Albeverio 1,2,∗, S. Mazzucchi 2 Institut für Angewandte Mathematik, Wegelerstr. 6, 53115 Bonn, Germany Received 25 May 2004; accepted 26 May 2004 Available online 6 August 2004 Abstract A general class of (finite dimensional) oscillatory integrals with polynomially growing phase func- tions is studied. A representation formula of the Parseval type is proven as well as a formula giving the integrals in terms of analytically continued absolutely convergent integrals. Their asymptotic ex- pansion for “strong oscillations” is given. The expansion is in powers of h¯ 1/2M,whereh¯ is a small parameters and 2M is the order of growth of the phase function. Additional assumptions on the integrands are found which are sufficient to yield convergent, resp. Borel summable, expansions. 2004 Elsevier SAS. All rights reserved. Résumé On étudie une classe générale d’intégrales oscillatoires en dimension finie avec une fonction de phase à croissance polynomiale. Une formule de représentation du type Parseval est démontrée, ainsi qu’une formule donnant les intégrales au moyen de la continuation analytique d’intégrales ab- solument convergentes. On donne les développements asymptotiques de ces intégrales dans le cas d’ “oscillations rapides”. Ces développements sont en puissance de h¯ 1/2M,oùh¯ est un petit para- mètre et 2M est l’ordre de croissance de la fonction de phase. Sous des conditions additionelles sur les integrands on obtient la convergence, resp. la sommabilité au sens de Borel, des développements asymptotiques. 2004 Elsevier SAS. All rights reserved. MSC: 28C05; 35S30; 34E05; 40G10; 81S40 * Corresponding author. E-mail addresses: [email protected] (S. Albeverio), [email protected] (S. Mazzucchi). 1 Cerfim (Locarno), Acc. Arch.(USI) (Mendrisio). 2 SFB611, BIBOS; IZKS; Dipartimento di Matematica, Università di Trento, 38050 Povo, Italia. 0007-4497/$ – see front matter 2004 Elsevier SAS. All rights reserved. doi:10.1016/j.bulsci.2004.05.005 2 S. Albeverio, S. Mazzucchi / Bull. Sci. math. 129 (2005) 1–23 Keywords: Oscillatory integrals; Asymptotic expansion; Stationary phase method 1. Introduction The study of finite dimensional oscillatory integrals of the form i Φ(x) e h¯ f(x)dx, (1) RN where h¯ is a non vanishing real parameter, Φ and f suitable real-valued smooth func- tions, is already a classical topic, largely developed in connections with various problems in mathematics and physics. Well known examples of simple integrals of the above form are the Fresnel integrals of the theory of wave diffraction and Airy’s integrals of the theory of rainbow. The theory of Fourier integral operators [25,26,34] also grew out of the investi- gation of oscillatory integrals. It allows the study of existence and regularity of a large class of elliptic and pseudoelliptic operators and provides constructive tools for the solutions of the corresponding equations. In particular one is interested in discussing the asymptotic behavior of the above integrals when the parameter h¯ goes to 0. The method of stationary phase provides a tool for such investigations and has many applications, such as the study of the classical limit of quantum mechanics (see [2,3,8,22,33,44]). In the general case of degenerate critical points of the phase function Φ, the theory of unfoldings of singularities is applied, see [13,20]. The extensions of the definition of oscillatory integrals to an infinite dimensional Hilbert space H and the implementation of a corresponding infinite-dimensional version of the stationary phase method has a particular interest in connection with the rigorous mathe- matical definition of the “Feynman path integrals”. Several methods has been discussed in literature, for instance by means of analytic continuation of Wiener integrals [16,17, 29,30,32,35,36,42,43], or by “infinite dimensional distributions” in the framework of the Hida calculus [19,24], via “complex Poisson measures” [1,34], via a “Laplace transform method” [5,31], or via a “Fourier transform approach”, see [3,4,6–8,21,27,28]. The latter method is particularly interesting as it is the only one by which a development of an in- finite dimensional stationary phase method has been performed. The phase functions that can be handled by this method are of the form “quadratic plus bounded perturbation”, that is Φ(x) =x,Tx+V(x),whereT is a self-adjoint operator and V is the Fourier transform of a complex bounded variation measure on H [3,8,40,41]. We also mention that the problem of definition and study of integrals of the form (1) but with h¯ ∈ C,Im(h)¯ < 0andΦ lower bounded has also been discussed. The convergence of the integral in this case is a simple matter, so the analysis has concentrated on a “pertur- bation theoretical” computation of the integral, like in [14,15], resp. on a Laplace method for handling the h¯ → 0 asymptotics, see, e.g. [3,9,11,12,38] (the latter method has some S. Albeverio, S. Mazzucchi / Bull. Sci. math. 129 (2005) 1–23 3 relations with the stationary phase method). The aim of the present paper is the definition of a “generalized Fresnel integral” of the form i Φ(x) e h¯ f(x)dx, (2) RN where Φ is a smooth function bounded at infinity by a polynomial P(x)on Rn,Im(h)¯ 0, h¯ = 0, and the study of the corresponding asymptotic expansion in powers of h¯ . The results we obtain will be generalized to the infinite dimensional case in a forthcoming paper [10] and applied to an extension of the class of phase functions for which the Feynman path integral had been defined before. In Section 2 we introduce the notations, recall some known results and prove the ex- istence of the oscillatory integral (2). In Section 3 we prove that when f belongs to a suitable class of functions, this generalized Fresnel integral can be explicitly computed by means of an absolutely convergent Lebesgue integral. We prove a representation formula of the Parseval type (Theorem 3) (similar to the one which was exploited in [7] in the case of quadratic phase functions), as well as a formula (Corollary 1 to Theorem 3) giving the integral in terms of analytically continued absolutely convergent integrals. Even if our main interest came from the case h¯ ∈ R \{0}, both formulae are valid for all h¯ ∈ C with Im(h)¯ 0, h¯ = 0. In the last section we consider the integral (2) in the particular case P(x)= A2M (x,...,x),whereA2M is a completely symmetric strictly positive covariant tensor of order 2M on RN , compute its detailed asymptotic power series expansion (in powers of h¯ 1/2M ,forIm(h)¯ 0, h¯ = 0) in the limit of “strong oscillations”, i.e. h¯ → 0. We give assumptions on the integrand f for having convergent, resp. Borel summable, expansions. 2. Definition of the generalized Fresnel integral Let us consider a finite dimensional real Hilbert space H,dim(H) = N,andletus N N identify it with R . We will denote its elements by x ∈ R , x = (x1,...,xN ). We recall the definition of oscillatory integrals proposed by Hörmander [25,26]. Definition 1. Let Φ be a continuous real-valued function on RN . The oscillatory integral on RN , with h¯ ∈ R \{0}, i Φ(x) e h¯ f(x)dx, RN is well defined if for each test function φ ∈ S(RN ), such that φ(0) = 1, the limit of the sequence of absolutely convergent integrals i Φ(x) lim e h¯ φ(εx)f (x) dx, ε↓0 RN 4 S. Albeverio, S. Mazzucchi / Bull. Sci. math. 129 (2005) 1–23 exists and is independent on φ. In this case the limit is denoted by i Φ(x) e h¯ f(x)dx. RN If the same holds only for φ such that φ(0) = 1andφ ∈ Σ, for some subset Σ of S(RN ), we say that the oscillatory integral exists in the Σ-sense and we shall denote it by the same symbol. Let us consider the space M(RN ) of complex bounded variation measures on RN en- dowed with the total variation norm. M(RN ) is a Banach algebra, where the product of two measures µ ∗ ν is by definition their convolution: µ ∗ ν(E) = µ(E − x)ν(dx), µ,ν ∈ M(RN ) RN and the unit element is the Dirac measure δ0. Let F(RN ) be the space of functions f : RN → C which are the Fourier transforms of N complex bounded variation measures µf ∈ M(R ): ik·x N f(x)= e µf (dk), µf ∈ M(R ). RN If there exists a self-adjoint linear isomorphism T : RN → RN such that the phase function Φ is given by Φ(x) =x,Tx and f ∈ F(RN ), then the oscillatory integral i h¯ x,Tx RN e f(x)dx can be explicitly computed by means of the following Parseval-type formula [3,21]: i x,Tx e 2h¯ f(x)dx RN − N/2 πi Ind(T ) −1/2 − ih¯ x,T −1x = (2πih)¯ e 2 det(T ) e 2 µf (dx), (3) RN where Ind(T ) is the number of negative eigenvalues of the operator T , counted with their multiplicity. In the following we will generalize the latter result to more general phase functions Φ, in particular those given by an even polynomial P(x)in the variables x1,...,xN : P(x)= A2M(x,...,x)+ A2M−1(x,...,x)+···+A1(x) + A0, (4) N where Ak are kth-order covariant tensors on R : RN × RN ×···×RN → R Ak : k-times S. Albeverio, S. Mazzucchi / Bull. Sci.
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