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Article

О методе ВКБ для разностных уравнений: вейлевский символ и фазовая геометрия

We study the asymptotics of solutions of linear difference equations (recurrence relations) with slowly varying coeffi cients. It is known that the local asymptotic behavior of solutions can be obtained similarly to the WKB approximation for linear differential equations. In contrast to the continuous case, one of the major obstacles to the widespread use of discrete WKB method is the lack of a geometric interpretation of the obtained asymptotic formulas. We show that it is possible to build a simple geometric interpretation of discrete WKB method if one consider the difference equation as pseudo-differential with corresponding Weyl symbol (Hamiltonian). We obtain such a geometric interpretation for local asymptotics, turning points, the Bohr-Sommerfeld rule and other basic elements of the semiclassical approximation.