Several aspects of the general and constructive spectral theory of quasiperiodic Schrödinger operators in one dimension are discussed. An explicit formula for the absolutely continuous (a.c.) spectral densities that yields an immediate proof of the fact that the Kolmogorov–Arnold–Moser (KAM) spectrum constructed by Dinaburg, Sinai, and Rüssmann [Funkt. Anal. Prilozen. 9, 8 (1975); Ann. Acad. Sci. 357, 90 (1980)] is a subset of the a.c. spectrum is provided. Some quasiperiodicity properties of the Deift–Simon a.c. eigenfunctions are proved, namely, that the normalized phase of such eigenfunctions is a quasiperiodic distribution. In the constructive part the Dinaburg–Sinai–Rüssmann theory is extended to quasiperiodic perturbations of periodic Schrödinger operators using a KAM Hamiltonian formalism based on a new treatment of perturbations of harmonic oscillators. Particular attention is devoted to the dependence upon the eigenvalue parameter and a complete control of KAM objects is achieved using the notion of Whitney smoothness.
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1987
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