On applications of Maupertuis-Jacobi correspondence for Hamiltonians $F(x,|p|)$ in some 2-D stationary semiclassical problems
S.Dobrokhotov, D.Minenkov (A.Ishlinskii Institute for Problems in, Mechanics Russian Academy of Sciences, Moscow institute of Physics and, Technology, Moscow, Russia), M.Rouleux (Aix-Marseille Universit\'e, CNRS,, CPT, UMR 7332, 13288 Marseille, France, Universit\'e de Toulon
TL;DR
This paper explores how the Maupertuis-Jacobi correspondence can simplify semiclassical asymptotic formulas for specific 2-D Hamiltonian systems, with applications to quantum mechanics, graphene physics, and water waves.
Contribution
It introduces a method to leverage classical mechanics principles to improve semiclassical analysis of Hamiltonians of the form F(x,|p|).
Findings
Simplifies 2-D asymptotic formulas using Maupertuis-Jacobi correspondence.
Applies the method to Schrödinger, Dirac, and water wave Hamiltonians.
Provides a unified approach for phase flow invariant Lagrangian manifolds.
Abstract
We make use of the Maupertuis -- Jacobi correspondence, well known in Classical Mechanics, to simplify 2-D asymptotic formulas based on Maslov's canonical operator, when constructing Lagrangian manifolds invariant with respect to phase flows for Hamiltonians of the form . As examples we consider Hamiltonians coming from the Schr\"odinger equation, the 2-D Dirac equation for graphene and linear water wave theory.
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