an:04046164
Zbl 0642.34018
Chierchia, Luigi
Absolutely continuous spectra of quasiperiodic Schr??dinger operators
EN
J. Math. Phys. 28, 2891-2898 (1987).
00152806
1987
j
34L99 35J10 37C55 34K99 47A10
spectral theory of one-dimensional Schr??dinger operators; spectral densities; Bloch waves; KAM objects
This paper is concerned with the spectral theory of one-dimensional Schr??dinger operators \(L\equiv -d^ 2/dx^ 2+v(x),\) where v is a (real) quasiperiodic function. The eigenvalue problem is discussed both from an abstract and a constructive point of view. For example, a general 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 \textit{E. I. Dinaburg}, \textit{Ja. G. Sinai} [Funkt. Anal. Prilozen. 9, 8-21 (1975; Zbl 0333.34014) and \textit{H. R??ssmann}, Nonlinear dynamics, int. Conf., New York 1979, Ann., N. Y. Acad. Sci. 357, 90-107 (1980; Zbl 0477.34007)] is a subset of the a.c. spectrum, is derived. Also, it is shown that the a.c. (generalized) eigenfunctions are ``weak'' Bloch waves, generalizing, in this sense, Floquet theory to the a.c. part of the spectrum of L. The problem of constructing explicitely smooth Bloch waves is then considered and the Dinaburg-Sinai-R??ssmann theory is extended to quasiperiodic perturbations of periodic Schr??dinger operators. The existence of such Bloch waves is shown to be intimately related to the canonical integrability of \(a(d+1)\)-dimensional (d\(\equiv \#\) of basic frequencies of v) classical Hamiltonian system parametrized by the eigenvalue E. Particular attention is devoted to the dependence upon E and a complete control of KAM objects is achieved using the notion of Whitney smoothness [\textit{H. Whitney}, Trans. Am. Math. Soc. 36, 63-89 (1934; Zbl 0008.24902)].
L.Chierchia
Zbl 0333.34014; Zbl 0477.34007; Zbl 0008.24902