Isotropic Tori, Complex Germ and Maslov Index, Normal Forms and Quasimodes of Multidimensional Spectral Problems

More than twenty years ago V. P. Maslov posed the question under what conditions it is possible to assign to invariant isotropic lower-dimensional tori of Hamiltonian systems sequences of asymptotic eigenvalues and eigenfunctions (spectral series) of the corresponding quantum mechanical and wave operators. In the present paper this question is answered in terms of the quadratic approximation to the theory of normal forms. We also discuss the quantization conditions for isotropic tori and their relation to topological, geometric, and dynamical characteristics (Maslov indices, rotation (winding) numbers, eigenvalues of dynamical flows, etc.).