L² stability and boundedness of the Fourier integral operators applied to the theory of the Feynman path integral
by
Wataru Ichinose
Shinshu University, Japan

Let x ∈ Rⁿ, [x\dot] ∈ Rⁿ and 0 ≤ t ≤ T, where T > 0 is arbitrary. We consider the Lagrangian function L(t, x, [x\dot]): = m|[x\dot]|²/2 + [x\dot]·A(t, x) - V(t, x), where m > 0 is a mass and V ∈ R and A = (A₁, ..., A_n) ∈ Rⁿ are electromagnetic potetntials. Let S(t, s;q) (0 ≤ s < t ≤ T) denote the classical action for a path q:[s, t]→ Rⁿ and p(x, w) be infinitely differentiable functions in R²ⁿ whose all derivatives are bounded. In this talk Fourier integral operators

P(t, s)f(x) : = Ö   m/(2pi(t - s))   n ó õ æ è expiS(t, s;qt, sx, y) ö ø p æ è x, (x-y)/ Ö   (t-s)   ö ø f(y)dy

are studied, where q^{t, s}_{x, y}(q) denotes the line y + (x -y)(q- s)/(t - s) (s ≤ q ≤ t).

Let ∥·∥ denote the L2 norm. Then the following are proved under some assumptions w.r.t. electromagnetic fields E(t, x) = -∂A/∂t - ÑV ∈ Rⁿ and B_jk(t, x) = ∂A_k/∂x_j - ∂A_j/∂x_k (j, k = 1, ..., n). There exists a r^* > 0 such that

∥P(t, s)f∥ ≤ C ∥f ∥,  0 < t-s ≤ r*

for all f ∈ L2 with a constant C ≥ 0 independent of t and s. In addtion, when p(x, w) = 1 is taken, then the stability, i.e.

∥P(t, s)f∥ ≤ eK(t-s) ∥f ∥,  0 < t-s ≤ r*

for all f ∈ L2 holds, where a constant K ≥ 0 is independent of t and s. These results are applied to prove the existence of the Feynman path integral defined by the time-slicing method through broken line paths.

References
[1] W. Ichinose, Commun. Math. Phys. 189(1997), 17-33.
[2] W. Ichinose, Rev. Math. Phys. 11(1999), 1001-1025.
[3] W. Ichinose, J. Math. Soc. Japan 55(2003), 957-983.

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Date received: June 13, 2005

Copyright © 2005 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # carf-30.