On applications of Maupertuis-Jacobi correspondence for Hamiltonians $F(x,\mathrm{\mid }p\mathrm{\mid })$ in some 2-D stationary semiclassical problems

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 $F(x,\mathrm{\mid }p\mathrm{\mid })$. As examples we consider Hamiltonians coming from the Schr"odinger equation, the 2-D Dirac equation for graphene and linear water wave theory.