Statement of Main Results

Abstract

Although the time slicing approximation of Feynman path integrals does not converge absolutely, it has a definite finite value if the potential satisfies Assumption 2.1 and if the time interval is short, because it is an oscillatory integral that satisfies Assumption 3.1. Furthermore, the time slicing approximation of Feynman path integrals converges to a limit as $|\varDelta |\rightarrow 0$. The limit turns out to be the fundamental solution of the Schrödinger equation. The semi-classical asymptotic formula called Birkhoff’s formula is proved from the standpoint of oscillatory integrals. In this chapter, these results as well as others are explained. Proofs will be given in subsequent chapters.