In the book, new methods in the theory of differential equations on manifolds with singularities are presented. The semiclassical theory in quantum mechanics is employed, adapted to operators that are degenerate in a typical way. The degeneracies may be induced by singular geometries, e.g., conical or cuspidal ones. A large variety of non-standard degenerate operators are also discussed.
The semiclassical approach yields new results and unexpected effects, also in classical situations. For instance, full asymptotic expansions for cuspidal singularities are constructed, and nonstationary problems on singular manifolds are treated. Moreover, finiteness theorems are obtained by using operator algebra methods in a unified framework. Finally the method of characteristics for general elliptic equations on manifolds with singularities is developed in the book.
