References
Aronszajn, N., On a problem of Weyl in the theory of Sturm-Liouville equations.Amer. J. Math., 79 (1957), 597–610.
Aubry, S., The new concept of transitions by breaking of analyticity in a crystallographic modle, inSolitons and Condensed Matter Physics (Oxford, 1978), pp. 264–277. Springer Ser. Solid-State Sci., 8. Springer-Verlag, Berlin-New York, 1978.
Aubry, S. &Andre, G., Analyticity breaking and Anderson localization in incommensurate lattices.Ann. Israel Phys. Soc., 3 (1980), 133–140.
Avron, J. &Simon, B., Singular continuous spectrum for a class of almost periodic Jacobi matrices.Bull. Amer. Math. Soc., 6 (1982), 81–85.
—, Almost periodic Schrödinger operators, II. The integrated density of states.Duke Math. J., 50 (1983), 369–391.
Bellissard, J., Lima, R. &Testard, D., A metal-insulator transition for the almost Mathieu model.Comm. Math. Phys., 88 (1983), 207–234.
Belokolos, E. D., A quantum particle in a one-dimensional deformed lattice. Estimates of lacunae dimension in the spectrum.Teoret. Mat. Fiz., 25 (1975), 344–57 (Russian).
Chojnacki, W., A generalized spectral duality theorem.Comm. Math. Phys., 143 (1992), 527–544.
Chulaevsky, V. &Delyon, F., Purely absolutely continuous spectrum for almost Mathieu operators.J. Statist. Phys., 55 (1989), 1279–1284.
Deift, P. &Simon, B., Almost periodic Schrödinger operators, III. The absolutely continuous spectrum in one dimension.Comm. Math. Phys., 90 (1983), 389–411.
Delyon, F., Absence of localization for the almost Mathieu equation.J. Phys. A., 20 (1987), L21-L23.
Dinaburg, E. &Sinai, YA., The one-dimensional Schrödinger equation with a quasiperiodic potential.Functional Anal. Appl., 9, (1975), 279–289.
Donoghue, W., On the perturbation of the spectra.Comm. Pure Appl. Math., 18 (1965), 559–579.
Fröhlich, J., Spencer, T. &Wittwer, P., Localization for a class of one-dimensional quasi-periodic Schrödinger operators.Comm. Math. Phys., 132 (1990), 5–25.
Gesztesy, F. &Simon, B., The xi function.Acta Math., 176 (1996), 49–71.
Goldstein, M., Laplace transform method in perturbation theory of the spectrum of Schrödinger operators, II. One-dimensional quasi-periodic potentials. Preprint, 1992.
Helffer, B. &Sjöstrand, J., Semi-classical analysis for Harper's equation, III. Cantor structure of the spectrum.Mém. Soc. Math., France (N.S.), 39 (1989), 1–139.
Jitomirskaya, S., Anderson localization for the almost Mathieu equation: A nonperturbative proof.Comm. Math. Phys., 165 (1994), 49–57.
—, Almost everything about the almost Mathieu operator, II, inXIth International Congress of Mathematical Physics (Paris, 1994), pp. 373–382. Internat. Press. Cambridge, MA, 1995.
Jitomirskaya, S. &Simon, B., Operators with singular continuous spectrum, III. Almost periodic Schrödinger operators.Comm. Math. Phys., 165 (1994), 201–205.
Last, Y., A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants.Comm. Math. Phys., 151 (1993), 183–192.
—, Zero measure for the almost Mathieu operator.Comm. Math. Phys., 164 (1994), 421–432.
—, Almost everything about the almost Mathieu operator, I, inXIth International Congress of Mathematical Physics (Paris, 1994), pp. 366–372. Internat. Press, Cambridge, MA, 1995.
Last, Y. & Simon, B., Eigenfunctions, transfer matrices, and a.c. spectrum of one-dimensional Schrödinger operators. Preprint.
Mandelshtam, V. &Zhitomirskaya, S., 1D-quasiperiodic operators. Latent symmetries.Comm. Math. Phys., 139 (1991), 589–604.
Simon, B., Spectral analysis of rank one perturbations and applications, inMathematical Quantum Theory, II.Schrödinger Operators (Vancouver BC, 1993), pp. 109–149. CRM Proc. Lecture Notes, 8. Amer. Math. Soc., Providence, RI, 1995.
Sinai, YA., Anderson localization for one-dimensional Schrödinger operator with quasiperiodic potential.J. Statist. Phys., 46 (1987), 861–909.
Author information
Authors and Affiliations
Additional information
This material is based upon work supported by the National Science Foundation under Grants DMS-9208029, DMS-9501265 and DMS-9401491. The Government has certain rights in this material.
Rights and permissions
About this article
Cite this article
Gordon, A.Y., Jitomirskaya, S., Last, Y. et al. Duality and singular continuous spectrum in the almost Mathieu equation. Acta Math. 178, 169–183 (1997). https://doi.org/10.1007/BF02392693
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02392693