This book presents Maslov's canonical operator method for finding asymptotic solutions of pseudo differential equations. The classical WKB method, so named in honor of its authors: Wentzel, Kramers and Brillouin, was created for finding quasi classical approximations in quantum mechanics. The simplicity, obviousness and "physicalness" of this method quickly made it popular: specialists in mathematical physics accepted it unequivocally as one of the weapons in their arsenal. The number of publications which are connected with the WKB method in one way or another can probably no longer be counted. The alternative name of the WKB method in diffraction problem- the ray method or the method of geometric optics - indicates that the approximations in the WKB method are constructed by means of rays. More precisely, the first approximation of the WKB method is constructed by means of rays (isolating the singular part), after which the usual methods of the (regular) theory of perturbations are applied. However, the ray method is not applicable at the points of space where the rays focus or form a caustic. Mathematically this fact expresses itself in the fact that the amplitude of the waves at such points become infinite.

Autorentext
Dana Mackenzie holds a doctorate in mathematics from Princeton University. After teaching mathematics at Duke University and Kenyon College for more than a decade, in 1997 he completed the Science Communication Program at the University of California at Santa Cruz. Since then he has been a freelance science writer, with articles appearing in such magazines as Science, Discover, American Scientist, Astronomy, and New Scientist. He lives in Santa Cruz with his wife as well as three cats and a dog

Klappentext

This book presents the topological and analytical foundations of the theory of Maslov's canonical operator for finding asymptotic solutions of a wide class of pseudodifferential equations. The topology and geometry of Lagrangian manifolds are studied in detail and the connections between Fourier integral operators and canonical operators established. Applications are proposed for the asymptotic solutions to the Cauchy problem and for the asymptotics of the spectra of non-self-dual operators. The authors set out to make more accessible to a wider readership - of specialists in topology, differential equations and functional analysis, including research students - the ideas of Maslov which were much more difficult and opaque in their original published form (1965). The book has been updated, as compared with the Russian edition, by numerous revisions within the text and the addition of two new chapters on more recent work.

Inhalt
I. The Topology of Lagrangian Manifolds.- 1. Some Topological Considerations.- 2. The Geometry of Real Lagrangian Manifolds.- 3. Complex Lagrangian Manifolds.- II. Maslov's Canonical Operator on a Real Lagrangian Manifold.- 4. Maslov's Canonical Operator (Real Case).- 5. The Asymptotics of Integrals of Rapidly Oscillating Functions with a Complex Phase.- 6. Maslov's Canonical Operator (Complex Case).- 7. Some Applications.- Appendix. Fourier-Maslov Integral Operators (The Smooth Theory of Maslov's Canonical Operator).- Notation Index.