Feynman Path Integral and Schrödinger Equation

Abstract

In this chapter we prove Feynman’s conjecture that the integral transform defined by the Feynman path integral is in fact the fundamental solution of the Schrödinger equation. The main tool is the $L^2$-boundedness theorem proof of which is left to Chap. 8 in Part II. By the way we shall prove that the main term of the semi-classical asymptotic of the fundamental solution of the Schrödinger equation satisfies the transport equations. At the end we obtain the second term of the semi-classical asymptotic and prove that it satisfies the second transport equation. Our discussion of this is different from the usual method originated by Birkhoff (Bull Am Math Soc 39:681–700 (1933) [11]). Our method enables us to obtain the bound of the remainder term.