The Complex WKB Method for Nonlinear Equations I, Softcover reprint of the original 1st ed. 1994 Linear Theory Progress in Mathematical Physics Series, Vol. 16

Introduction. Equations and problems of narrow beam mechanics. Hamiltonian formalism of narrow beams. Approximate solutions of the nonstationary transport equation. Stationary Hamilton-Jacobi and transport equations. Complex Hamiltonian formalism of compact (cyclic) beams. Canonical operators on Lagrangian manifolds with complex germ and their applications to spectral problems of quantum mechanics.

I. Equations and problems of narrow beam mechanics.- II. Hamiltonian formalism of narrow beams.- III. Approximate solutions of the nonstationary transport equation.- IV. Stationary Hamilton-Jacobi and transport equations.- V. Complex Hamiltonian formalism of compact (cyclic) beams.- VI. Canonical operators on Lagrangian manifolds with complex germ and their applications to spectral problems of quantum mechanics.- References.- Appendix A Complex germ generated by a linear connection.- Appendix B Asymptotic solutions with pure imaginary phase and the tunnel equation.- Appendix C Analytic asymptotics of oscillatory decreasing type (heuristic considerations).

The central mathematical idea is to use topological methods to analyse isotropic invariant manifolds in order to obtain asymptotic series of eigenvalues and eigenvectors for the multidimensional Schrödinger equation, and also to take into account the so-called tunnel effects. This English edition includes new examples of applications to physics, hitherto unpublished or available only in Russian.