Feynman path integrals for polynomially growing potentials

Abstract

A general class of infinite dimensional oscillatory integrals with polynomially growing phase functions is studied. A representation formula of the Parseval type is proved, as well as a formula giving the integrals in terms of analytically continued absolutely convergent integrals. These results are applied to provide a rigorous Feynman path integral representation for the solution of the time-dependent Schrödinger equation with a quartic anharmonic potential. The Borel summability of the asymptotic expansion of the solution in power series of the coupling constant is also proved.

Theory and Applications of Infinite Dimensional Oscillatory Integrals

“…of formula (19) extends to an analytic function of the variable λ, for Im(λ) > 0, and is still continuous for Im(λ) = 0 [20,21]. The leading idea of the proof is the computation of the .…”

Schwartz operators

“…According to theorem 8 one can see that the functional (I , D(I )) is a continuous extension of the functional (37). This enlargement of the class of integrable function and the connection of the functional (37) with the solution of the Schrödinger equation (on R d with potential consisting of a harmonic part plus a part V ∈ F (R d )) has been exploited in [21] for an extension of theorem 7 to the case of potentials V with polynomial growth. In the latter case one obtains a representation in terms of expectations od complex-valued functions with respect to the gauusian measure associated to the abstract Wiener space built on the Hilbert space H.…”

“…This program has been initiated by Yu. Daletskii [57,58] and K. Itô [93], and continued systematically in [14,13,68,4,2,3,5,8,9,10,11,15,16,18,19,20,21,22,23,24,25,26,27,30,34,47,53,62,68,69,72,78,80,91,96,97,100,101,106,111,113,121,130,144,151], see also [1,28] and references therein. A connection with certain infinite dimensional distributions has also been achieved …”

Functional-Integral Solution for the Schrödinger Equation With Polynomial Potential: A White Noise Approach

“…Then it is possible to prove (see [4,6]), that the states φ ∈ S 2 , ψ ∈ S 1 and the time dependent potential V satisfy the conditions 1, 2 and 3.…”

“…Nevertheless some partial results have been obtained in [4,5,25] concerning the representation of the solution of the Schrödinger equation with a quartic oscillator potential in terms of infinite dimensional oscillatory integrals. An analytically-continued Wiener integral representation for the weak solution of the Schrödinger equation (in the sense of equation (13)) has been proposed.…”