Remarks on Fujiwara’s stationary phase method on a space of large dimension with a phase function involving electromagnetic fields

Abstract

We consider an oscillatory integral of the formHere each xj, j = 0, 1,…, L, runs in Rd, ν > 1 is a constant and tj, j = 1,…,L, are positive constants. Fujiwara [5] discussed this integral for L large and developed the stationary phase method with an estimate of the remainder term for the phase function S(xL,…, x0) coming from the action integral for a particle in an electric field. But his results cannot be applied to the integral which naturally arises in the discussion of quantum mechanics of a charged particle moving in a magnetic field. In this paper we extend his results to the case for the phase function involving both electric and magnetic fields.