SEMICLASSICAL MASLOV ASYMPTOTICS WITH COMPLEX PHASES .1. 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 int egrable and classically nonintegrable Hamiltonian systems. In the firs t case, the systems admit families of invariant Lagrangian tori (of co mplete dimension equal to the dimension n of the configuration space) whose quantization in accordance with the Bohr-Sommerfeld rule with al lowance for the Maslov index gives the semiclassical series in the reg ion of large quantum numbers. In the nonintegrable case, families of L agrangian tori with complete dimension do not exist. However, in the r egion of regular (nonchaotic) motion, such systems do have invariant L agrangian tori of dimension k < n (incomplete dimension). The construc tion method associates the families of such tori with spectral series covering the region of ''intermediate'' quantum numbers. The construct ion includes, in particular, new quantization conditions of Bohr-Somme rfeld type in which other characteristics of the tori appear instead o f the Maslov index. Applications and also generalizations of the theor y to Lie groups will be presented in subsequent publications of the se ries.