QUASI-CLASSICAL SPECTRAL SERIES OF THE DIRAC OPERATORS CORRESPONDING TO QUANTIZED 2-DIMENSIONAL LAGRANGIAN TORI

Based on the Maslov complex germ theory, a method of constructing the quasiclassical spectral series for the Dirac operator is proposed. The case when the corresponding relativistic Hamiltonian system is non-integrable and it admits a family of invariant two-dimensional stable Lagrangian tori containing the focal points is considered. The resulting quantization conditions for the above family generalize the Bohr-Sommerfeld-Maslov conditions and include new additional characteristics. The quasi-classical asymptotics obtained are regular over the full classically allowed domain. They also form an asymptotically complete and orthonormal set. Examples which use the proposed technique of the quasi-classical quantization are analysed.