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The method of Stationary phase The Legendre Transforma
The HamiltonJacobi Equation The Hamilton System
11 other sections not shown
admits a diffeomorphic asymptotic formulae bicharacteristic canonical atlas canonical chart canonical operator canonical transformation Cauchy data Cauchy problem caustic chart Q classical mechanics coefficients commutation formula compact support Consequently construct corresponding curve cycle of singularities definition denote derivative diffeomorphic projection differential equations differential operators Dirac equation domain eigenfunctions eigenvalues eikonal equation equal Equation 0.3 estimate exp iAS(x f.a. solution fixed focal coordinates follows function S(x given Green matrix Hamilton system Hamilton-Jacobi equation Hamiltonian inerdex infinitely differentiable integral introduce Jacobian Laplace-Beltrami operator leading term Legendre transformation Lemma Let the assumptions matrix G Morse index multi-indices neighbourhood of point obtain p-representation p,dx parameter partition of unity phase space phase trajectories precanonical operator Proof PROPOSITION real-valued satisfied Schrodinger equation small neighbourhood smooth solution of Equation stationary phase method stationary point symplectic Theorem 2.6 transport equation tube of trajectories variables vector function volume element zero