Abstract
This paper provides an elementary proof of the classical limit of the Schrödinger equation with WKB type initial data and over arbitrary long finite time intervals. We use only the stationary phase method and the Laptev-Sigalsimple and elegant construction of a parametrix for Schrödinger 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 obtainedwhen using the Laptev-Sigal parametrix are related to the Maslov index.
Highlights
This paper provides an elementary proof of the classical limit of the Schrodinger equation with WKB type initial data and over arbitrary long finite time intervals
The classical limit of quantum mechanics is defined by the scaling assumption 2π ≪ mL2/T — i.e. the typical action of the particle considered is large compared to
Before discussing the classical limit of the Schrodinger equation, we need to recall a few preliminary results on either the quantum dynamics defined by (3) or the classical dynamics in phase space associated to the Schrodinger equation (3)
Summary
V (x)ψ for the wave function ψ of a point particle of mass m subject to the action of an external potential V ≡ V (x) ∈ R. Our purpose in the present work is to provide a short and self-contained proof of this result, based on Laptev-Sigal lucid construction of a parametrix for the Schrodinger equation (1) in [12]. In this way, we avoid using either Maslov’s formalism [15, 16] of the canonical operator, or the semiclassical analogue of Hormander’s global theory [11] of Lagrangian distributions or Fourier integral operators. The Laptev-Sigal construction, which is is remarkably short (pp. 753–759 in [12]), uses instead Fourier integrals with a complex phase whose imaginary part is quadratic
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