The meaning of Maslov's asymptotic method: The need of Planck's constant in mathematics

Abstract

H. Poincaré defined asymptotic expansions. Their use by the W. K. B. method introduced a new kind of solution of linear differential equations. Maslov showed their singularities to be merely apparent. The clarification of those results leads to the introduction of "Lagrangian functions", of their scalar product and of "Lagrangian operators*', which constitutes a new structure: the "Lagrangian analysis*'. The last step of its definition requires the choice of a constant. That constant has to be Planck's constant, when the equation is the Schrodinger or the Dirac equation describing the hydrogen atom-the study of atoms with several electrons is very incomplete. © 1981 American Mathematical Society.