On the Classical Limit of the Schrödinger Equation

Claude Bardos (1) , François Golse (2) , Peter Markowich (3) , Thierry Paul (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 CMLS - Centre de Mathématiques Laurent Schwartz
3 MCSE Division

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-Sigal simple 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 obtained when using the Laptev-Sigal parametrix are related to the Maslov index.

Mots clés: Schrödinger equation; Classical limit; WKB expansion; Caustic; Fourier integral operators; Lagrangian manifold; Maslov index

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Claude Bardos, François Golse, Peter Markowich, Thierry Paul. On the Classical Limit of the Schrödinger Equation. Discrete and Continuous Dynamical Systems - Series A, 2015, 35 (12), pp.5689-5709. ⟨10.3934/dcds.2015.35.5689⟩. ⟨hal-01074071⟩