The Complex WKB Method for Nonlinear Equations I

Part 1 Equations and problems of narrow beam mechanics: asymptotic solutions of narrow beam type for partial differential equations with small parameter; systems of canonical equations; inequalities of the garding type; approximate solutions of the canonical system; generalized Cauchy problem and nonstationary transport equation. Part 2 Hamiltonian formalism of narrow beams: model problem; auxiliary facts from symplectic geometry of the phase space; Lagrangian manifolds with real germ; phase and action on Lagrangian manifolds with real germ; phase reconstruction; Lagrangian manifolds with complex germ; dissipation conditions; action on Lagrangian manifolds with complex germ; canonical transformations of Lagrangian manifolds with complex germ; approximate complex solutions of the nonstationary Hamilton-Jacobi equation. Part 3 Approximate solutions of the nonstationary transport equation: approximate real solutions of the transport equation; approximate complex solutions of the nonstationary transport equation; creation and annihilation operators for the generalized nonstationary transport equation; creation and annihilation operators - general case; the spaces of functions S([lambdak, gamman/T lambdak]); generalized transport equation with nonzero right-hand side. Part 4 Stationary Hamilton-Jacobi and transport equations: canonical system of stationary equations; invariant Lagrangian manifolds with complex germ; approximate solutions of the stationary Hamilton-Jacobi equation and the transport equation; the generalized Cauchy problem for stationary Hamilton-Jacobi equations; the Cauchy problem in the plane for transport equations; generalized stationary transport equation; examples; generalized eigenfunctions of the Helmholtz operator. Part 5 Complex Hamiltonian formalism of compact (cyclic) beams: setting the problem; invariant zero-dimensional Lagrangian manifolds with complex germ; approximate solutions of the generalized transport equation concentrated in the neighborhood of a point; family of closed curves with complex germ; functions on a family of closed curves with complex germ; creation operators; invariant closed curves with complex germ; approximate cyclic solutions of the stationary Hamilton-Jacobi equation; approximate solutions of the generalized transport equation. Part 6 Canonical operators on Lagrangian manifolds with complex germ and their applications to spectral problems of quantum mechanics: invariant closed curves with complex germ in systems with one cyclic variable; semiclassical spectral series for Schrodinger and Klein-Gordon operators in electromagnetic fields with axial symmetry corresponding to relative equilibrium positions; construction of the canonical operator on Lagrangian manifolds with complex germ; canonical operators and polynomial beams over isotropic manifolds; example; table of asymptotic spectral series.