## Almost Periodic Operators and Related Nonlinear Integrable SystemsDelves into contemporary advances in the mathematical theory of non-ideal crystalline structures - the spectral theory of differential and finite difference operators with almost periodic coefficients. Covers such topics as Cantor spectra, Anderson locali |

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### Contents

Basic facts of the theory of periodic operators | 3 |

The integrated density of states | 21 |

Methods of algebraic geometry | 43 |

The quasiperiodic operators | 76 |

100 | |

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### Common terms and phrases

absolute continuous algebraic approximation APSO Assume Aubry duality Bloch function Bloch phase boundary conditions Cantor set Cauchy Cauchy data Chapter coefficients consider const convergence Corollary corresponding decaying deduce defined density differential equation Dubrovin equations eigenfunctions eigenvalues endpoints equal ergodic estimate exists exponentially finite number finite-difference finite-zone potentials fixed Fourier frequency module Hence holomorphic independent induction inequality integrable kH(E kv(E L-periodic Laplace transform lattice Lebesgue measure Lemma limit periodic operator linear space LPSO Lyapunov index Marchenko-Ostrovskii mathematical Mathieu equation matrix monodromy operator nonlinear norm operator H periodic functions periodic potential perturbation point spectrum polynomial potential V(x proof pseudomonotonic pure point QPSO quasi-periodic operators quasimomentum resonance Riemannian surface satisfy Schrodinger equation self-adjoint self-adjoint operators sequence solution spectral bands spectral gaps spectral measure spectral problem spectral theory subspace surface F Theorem Thouless formula trace formulae uniformly bounded upper half-plane vector Wronskian zero