Lagrangian Manifolds and the Maslov Operator

ISBN-13: 9783642647659 / Angielski / Miękka / 2011 / 395 str.

Aleksandr S. Mishchenko; Viktor E. Shatalov; Boris Yu Sternin

Opis

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.

Spis treści

I. The Topology of Lagrangian Manifolds.- 1. Some Topological Considerations.- 1.1 Manifolds and Bundles.- 1.2 Theorems on Transversal Regularity.- 1.3 The Index of Intersection of Submanifolds.- 1.4 Homotopy Groups.- 2. The Geometry of Real Lagrangian Manifolds.- 2.1 Lagrangian Manifolds in Hamiltonian Space.- 2.2 The Cohomology of the Lagrangian Grassmannian.- 2.3 Characteristic Classes of Lagrangian Manifolds.- 2.4 Lagrangian Manifolds in General Position.- 3. Complex Lagrangian Manifolds.- 3.1 The Grassmannian of Positive Lagrangian Planes.- 3.2 The Maslov Index of Complex Lagrangian Manifolds.- 3.3 Analysis on s-Analytic Manifolds.- 3.4 Positive Lagrangian s-Analytic Manifolds.- II. Maslov’s Canonical Operator on a Real Lagrangian Manifold.- 4. Maslov’s Canonical Operator (Real Case).- 4.1 The Construction of Maslov’s Elementary Canonical Operator.- 4.2 Commutation of Maslov’s Canonical Operator and the Hamiltonian Operator.- 5. The Asymptotics of Integrals of Rapidly Oscillating Functions with a Complex Phase.- 5.1 The Formula for Asymptotic Expansion of the Integral of a Rapidly-Oscillating Function.- 5.2 Proof of Proposition 1.2.- 6. Maslov’s Canonical Operator (Complex Case).- 6.1 Maslov’s Elementary Operator on a Complex Lagrangian Manifold.- 6.2 Commutation of the Canonical Operator and the Hamiltonian (Elementary Theory).- 6.3 Commutation of Maslov’s Canonical Operator and the Hamiltonian (General Theory).- 6.4 Other Approaches.- 6.5 Appendix. The 1/h-Fourier Transform.- 7. Some Applications.- 7.1 Asymptotic Solutions of the Cauchy Problem.- 7.2 Asymptotics of the Spectrum of 1/h-Pseudodifferential Operators.- 7.3 Systems of Equations.- Appendix. Fourier-Maslov Integral Operators (The Smooth Theory of Maslov’s Canonical Operator).- Notation Index.

O autorze

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