Finite dimensional approximation approach to oscillatory integrals and stationary phase in infinite dimensions
Here it is presented a new approach to infinite dimensional oscillatory integrals and the corresponding method of stationary phase. This new approach extends in one direction the definition of the approach in Fourier transforms of the integral and the method of stationary phase for it. The infinite dimensional oscillatory integrals are defined in Section 2 as limits of finite dimensional approximations. In Sections 3, 4 and 5 there are discussed respectively the cases where there is a unique stationary point at the origin, the stationary point is not necessarily at the origin and where the stationary point is not unique. An extension to the case of a more general phase is also presented in Section 6. Moreover, two examples are given in Section 7. One of them provide a representation of the solution of the time dependent Schrödinger equation with a constant magnetic field in terms of infinite dimensional oscillatory integrals. The second one concerns a representation of Gibbs states in classical statistical mechanics by oscillatory integrals.
infinite dimensional oscillatory integrals
time dependent Schrödinger equation with a constant magnetic field
representation of Gibbs states in classical statistical mechanics
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