On the Classical Limit of the Schr\"{o}dinger Equation

TL;DR

This paper offers an elementary proof of the classical limit of the Schrödinger equation using WKB initial data, stationary phase, and the Laptev-Sigal parametrix, clarifying phase shifts and the Maslov index.

Contribution

It provides a simple, rigorous proof of the classical limit for Schrödinger equations with detailed analysis of phase shifts and caustics.

Findings

Elementary proof of classical limit using stationary phase

Explicit relation between phase shifts and Maslov index

Applicability over arbitrary long finite time intervals

Abstract

This paper provides an elementary proof of the classical limit of the Schr\"{o}dinger equation with WKB type initial data and over arbitrary long finite time intervals. We use only the stationary phase method and the Laptev-Sigal simple and elegant construction of a parametrix for Schr\"{o}dinger type equations [A. Laptev, I. Sigal, Review of Math. Phys. 12 (2000), 749-766]. We also explain in detail how the phase shifts across caustics obtained when using the Laptev-Sigal parametrix are related to the Maslov index.