Formal oscillatory distributions

Article type: Research Article

Authors: Karabegov, Alexander^{; *}

Affiliations: Department of Mathematics, Abilene Christian University, ACU Box 28012, Abilene, TX 79699-8012, USA. E-mail: [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: The formal asymptotic expansion of an oscillatory integral whose phase function has one nondegenerate critical point is a formal distribution supported at the critical point which is applied to the amplitude. This formal distribution is called a formal oscillatory integral (FOI). We introduce the notion of a formal oscillatory distribution supported at a point. We prove that a formal distribution is given by some FOI if and only if it is an oscillatory distribution that has a certain nondegeneracy property. We also prove that a star product ⋆ on a Poisson manifold M is natural in the sense of Gutt and Rawnsley if and only if the formal distribution f⊗g↦(f⋆g)(x) is oscillatory for every x∈M.

Keywords: Formal oscillatory integral, oscillatory distribution, natural deformation quantization

DOI: 10.3233/ASY-201662

Journal: Asymptotic Analysis, vol. 126, no. 1-2, pp. 45-63, 2022