Complex Maslov germs in abstract spaces

For various evolution equations for an element of a Hilbert space one uses different asymptotic methods to construct approximate solutions of these equations, which are expressed in terms of points (that are time-dependent and satisfy certain equations) in a smooth manifold ${\mathscr Y}$ and elements of a Hilbert space ${\mathscr F}_y$. In the present paper the properties of asymptotic solutions are studied under fairly general assumptions on the map associating a pair $y\in {\mathscr Y}$, $f\in {\mathscr F}_y$ with an asymptotic formula. An analogue of the concept of complex Maslov germ is introduced in the abstract case and its properties are studied. An analogue of the theory of Lagrangian manifolds with complex germ is discussed. The connection between the existence of an invariant complex germ and the stability of the solution of the equation for a point in the smooth manifold ${\mathscr Y}$ is investigated. The results so obtained can be used for the construction and geometric interpretation of new asymptotic solutions of evolution equations in the case when some class of asymptotic solutions is already known.