The Complex WKB Method for Nonlinear Equations I: Linear Theory

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The central mathematical idea of this text is to use topological methods to analyze isotropic invariant manifolds in order to obtain asymptotic series of eigenvalues and eigenvectors for the multi-dimensional Schroedinger equation, and also to take into account the so-called tunnel effects. Finite-dimensional linear theory is reviewed in detail. Infinite-dimensional linear theory and its applications to quantum statistics and quantum field theory, as well as the nonlinear theory (involving instantons), are considered in the second volume.