## Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation.(English)Zbl 0753.34055

Summary: We show that the one-dimensional Schrödinger equation with a quasi-periodic potential which is analytic on its hull admits a Floquet representation for almost every energy $$E$$ in the upper part of the spectrum. We prove that the upper part of the spectrum is purely absolutely continuous and that, for a generic potential, it is a Cantor set. We also show that for a small potential these results extend to the whole spectrum.

### MSC:

 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 34C25 Periodic solutions to ordinary differential equations 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
Full Text:

### References:

 [1] Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys.84, 403–438 (1982) · Zbl 0497.35026 [2] Moser, J., Pöschel, J.: An extension of a result by Dinaburg and Sinai on quasi-periodic potentials. Comment. Math. Helvetici59, 39–85 (1984) · Zbl 0533.34023 [3] Dinaburg, E. I., Sinai, Y. G.: The one dimensional Schrödinger equation with quasi-periodic potential. Funkt. Anal. i. Priloz.9, 8–21 (1975) · Zbl 0357.58011 [4] Rüssmann, H.: On the one dimensional Schrödinger equation with a quasi-periodic potential. Ann. N. Y. Acad. Sci.357, 90–107 (1980) [5] Deift, P., Simon, B.: Almost periodic Schrödinger operators. III. The absolutely continuous spectrum in one dimension. Commun. Math. Phys.90, 389–411 (1983) · Zbl 0562.35026 [6] Fröhlich, J., Spencer, T., Wittwer, P.: Localization for a class of one dimensional quasi-periodic Schrödinger operators. Commun. Math. Phys.132, 5–25 (1990) · Zbl 0722.34070 [7] Sinai, Ya. G.: Anderson localization for the one-dimensional difference Schrödinger operator with quasiperiodic potentials. J. Stat. Phys.46, 861–909 (1987) · Zbl 0682.34023 [8] Surace, S.: The Schrödinger equation with a quasi-periodic potential, Thesis N. Y. University · Zbl 0712.34094 [9] Spencer, T.: Ergodic Schrodinger operators. In P. Rabinoqitz, E. Zehnder (eds.). Analysis etc. Volume for J. Moser’s sixtieth birthday, New York Press 1989 [10] Kotani, S.: Lyapunov Indices determine absolutely continuous spectra of stationary random 1-dimensional Schrödinger operators. Proc. of Taniguchi Sympos., SA Katata, 225–247, (1982) [11] Chulaevsky, V. A., Delyon, F.: Purely absolutely continuous spectrum for almost Mathieu operators, preprint (1989) · Zbl 0714.34129 [12] Albanese, C.: Quasiperiodic Schrödinger operators with pure absolutely continuous spectrum, preprint Courant Institute, New York (1990) [13] Jörgens, K., Rellich, F.: Eigenwerttheorie gewöhnlicher Differentialgleichungen. Berlin, Heidelberg, New York: 1976 · Zbl 0347.34013 [14] Dym, H., McKean, H. P.: Gaussian Processes, Function Theory, and the Inverse spectral Problem. New York: Academic Press (1972) · Zbl 0242.42001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.