紹介
This monograph provides the first survey of Floquet theory for partial differential equations with periodic coefficients. The author investigates, among others, hypoelliptic, parabolic, elliptic and Schrodinger equations, and boundary value problems arising in applications. In particular, results are given about completeness of the set of Floquet solutions, Floquet expansions of arbitrary solutions, the distribution of Floquet exponents and quasimomentums, the solvability of nonhomogeneous equations, the existence of decreasing or bounded solutions, and the structure of the spectrum of periodic operators. Many of the results discussed here have been available only in research papers until now. The role of the Floquet-Lyapunov theory for ordinary differential equations is well known, but for partial differential equations an analog of this theory has been developed only recently. This theory is of great importance for the quantum theory of solids, the theory of wave guides, scattering theory and other fields of mathematical and theoretical physics. Some chapters devoted to operator theory may be of particular interest to specialists in complex analysis and functional analysis.
目次
1. Holomorphic Fredholm Operator Functions.- 1.1. Lifting and open mapping theorems.- 1.2. Some classes of linear operators.- 1.3. Banach vector bundles.- 1.4. Fredholm operators that depend continuously on a parameter.- 1.5. Some information from complex analysis.- A. Interpolation of entire functions of finite order.- B. Some information from the complex analysis in several variables.- C. Some problems of infinite-dimensional complex analysis.- 1.6. Fredholm operators that depend holomorphically on a parameter.- 1.7. Image and cokernel of a Fredholm morphism in spaces of holomorphic sections.- 1.8. Image and cokernel of a Fredholm morphism in spaces of holomorphic sections with bounds.- 1.9. Comments and references.- 2. Spaces, Operators and Transforms.- 2.1. Basic spaces and operators.- 2.2. Fourier transform on the group of periods.- 2.3. Comments and references.- 3. Floquet Theory for Hypoelliptic Equations and Systems in the Whole Space.- 3.1. Floquet - Bloch solutions. Quasimomentums and Floquet exponents.- 3.2. Floquet expansion of solutions of exponential growth.- 3.3. Completeness of Floquet solutions in a class of solutions of faster growth.- 3.4. Other classes of equations.- A. Elliptic systems.- B. Hypoelliptic equations and systems.- C. Pseudodifferential equations.- D. Smoothness of coefficients.- 3.5. Comments and references.- 4. Properties of Solutions of Periodic Equations.- 4.1. Distribution of quasimomentums and decreasing solutions.- 4.2. Solvability of non-homogeneous equations.- 4.3. Bloch property.- 4.4. Quasimomentum dispersion relation. Bloch variety.- 4.5. Some problems of spectral theory.- 4.6. Positive solutions.- 4.7. Comments and references.- 5. Evolution Equations.- 5.1. Abstract hypoelliptic evolution equations on the whole axis.- 5.2. Some degenerate cases.- 5.3. Cauchy problem for abstract parabolic equations.- 5.4. Elliptic and parabolic boundary value problems in a cylinder.- A. Elliptic problems.- B. Parabolic problems.- 5.5. Comments and references.- 6. Other Classes of Problems.- 6.1. Equations with deviating arguments.- 6.2. Equations with coefficients that do not depend on some arguments.- 6.3. Invariant differential equations on Riemannian symmetric spaces of non-compact type.- 6.4. Comments and references.- Index of symbols.