## Floquet Theory for Partial Differential EquationsLinear differential equations with periodic coefficients constitute a well developed part of the theory of ordinary differential equations [17, 94, 156, 177, 178, 272, 389]. They arise in many physical and technical applications [177, 178, 272]. A new wave of interest in this subject has been stimulated during the last two decades by the development of the inverse scattering method for integration of nonlinear differential equations. This has led to significant progress in this traditional area [27, 71, 72, 111 119, 250, 276, 277, 284, 286, 287, 312, 313, 337, 349, 354, 392, 393, 403, 404]. At the same time, many theoretical and applied problems lead to periodic partial differential equations. We can mention, for instance, quantum mechanics [14, 18, 40, 54, 60, 91, 92, 107, 123, 157-160, 192, 193, 204, 315, 367, 412, 414, 415, 417], hydrodynamics [179, 180], elasticity theory [395], the theory of guided waves [87-89, 208, 300], homogenization theory [29, 41, 348], direct and inverse scattering [175, 206, 216, 314, 388, 406-408], parametric resonance theory [122, 178], and spectral theory and spectral geometry [103 105, 381, 382, 389]. There is a sjgnificant distinction between the cases of ordinary and partial differential periodic equations. The main tool of the theory of periodic ordinary differential equations is the so-called Floquet theory [17, 94, 120, 156, 177, 267, 272, 389]. Its central result is the following theorem (sometimes called Floquet-Lyapunov theorem) [120, 267]. |

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### Contents

HOLOMORPHIC FREDHOLM OPERATOR FUNCTIONS | 1 |

12 Some classes of linear operators | 5 |

13 Banach vector bundles | 21 |

14 Fredholm operators that depend continuously on a parameter | 28 |

15 Some information from complex analysis | 32 |

A Interpolation of entire functions of finite order | 34 |

B Some information from the complex analysis in several variables | 54 |

C Some problems of infinitedimensional complex analysis | 60 |

35 Comments and references | 121 |

PROPERTIES OF SOLUTIONS OF PERIODIC EQUATIONS | 125 |

42 Solvability of nonhomogeneous equations | 147 |

43 Bloch property | 150 |

44 Quasimomentum dispersion relation Bloch variety | 151 |

45 Some problems of spectral theory | 158 |

46 Positive solutions | 172 |

47 Comments and references | 183 |

16 Fredholm operators that depend holomorphically on a parameter | 66 |

17 Image and cokernel of a Fredholm morphism in spaces of holomorphic sections | 74 |

18 Image and cokernel of a Fredholm morphism in spaces of holomorphic sections with bounds | 76 |

19 Comments and references | 86 |

SPACES OPERATORS AND TRANSFORMATIONS | 91 |

22 Fourier transform on the group of periods | 93 |

23 Comments and references | 102 |

FLOQUET THEORY FOR HYPOELLIPTIC EQUATIONS AND SYSTEMS IN THE WHOLE SPACE | 103 |

32 Floquet expansions of solutions of exponential growth | 110 |

33 Completeness of Floquet solutions in a class of solutions of faster growth | 112 |

34 Other classes of equations | 115 |

B Hypoelliptic equations and systems | 116 |

C Pseudodifferential equation Let LxD be a pseudodifferential operator of | 117 |

D Smoothness of coefficients | 118 |

EVOLUTION EQUATIONS | 187 |

52 Some degenerate cases | 209 |

53 Cauchy problem for abstract parabolic equations | 224 |

54 Elliptic and parabolic boundary value problems in a cylinder | 244 |

B Parabolic Problems | 255 |

55 Comments and references | 259 |

OTHER CLASSES OF PROBLEMS | 263 |

62 Equations with coefficients that do not depend on some variables | 268 |

63 Invariant differential equations on Riemannian symmetric spaces of noncompact type | 283 |

64 Comments and references | 297 |

Bibliography | 303 |

345 | |

349 | |

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1-periodic algebra analogous analytic subset assume Banach bundles Banach space belongs Bloch solution Bloch variety boundary bounded chapter cokernel compact complex constant coefficient convergence corollary corresponding define differential operators dim Ker domain Due to theorem eigenvalue elliptic equations elliptic operators embedding English transl entire function equivalent estimate exists exponential fiber Floquet exponents Floquet solutions Floquet theory Fourier transform Frechet space Fredholm morphism Fredholm operator Hence Hilbert space holomorphic functions hypoelliptic invertible isomorphism kernel lemma Let us consider Let us denote linear operators mapping Math matrix morphism multiplication neighborhood non-zero norm obtain obviously operator function parabolic partial differential equations periodic coefficients perturbation polynomial potential proof of theorem proved quasimomentums respect Russian satisfies scalar Schrodinger operator sequence sheaf smooth solvability spectrum statement Stein manifold subbundle subspace surjective topology trivial bundle values vector bundle zero

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Page 321 - Geometrical background for the perturbation theory of the polyharmonic operator with periodic potentials.