Semiclassical maslov asymptotics with complex phases. I. General approach

A method of constructing semiclassical asymptotics with complex phases is presented for multidimensional spectral problems (scalar, vector, and with operator-valued symbol) corresponding to both classically integrable and classically nonintegrable Hamiltonian systems. In the first case, the systems admit families of invariant Lagrangian tori (of complete dimension equal to the dimensionn of the configuration space) whose quantization in accordance with the Bohr–Sommerfeld rule with allowance for the Maslov index gives the semiclassical series in the region of large quantum numbers. In the nonintegrable case, families of Lagrangian tori with complete dimension do not exist. However, in the region of regular (nonchaotic) motion, such systems do have invariant Lagrangian tori of dimensionk (incomplete dimension). The construction method associates the families of such tori with spectral series covering the region of intermediate quantum numbers. The construction includes, in particular, new quantization conditions of Bohr–Sommerfeld type in which other characteristics of the tori appear instead of the Maslov index. Applications and also generalizations of the theory to Lie groups will be presented in subsequent publications of the series.