Acta Mathematica

Duality and singular continuous spectrum in the almost Mathieu equation

A. Y. Gordon, S. Jitomirskaya, Y. Last, and B. Simon

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Note

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.

Article information

Source
Acta Math. Volume 178, Number 2 (1997), 169-183.

Dates
Received: 7 February 1996
First available in Project Euclid: 31 January 2017

Permanent link to this document
http://projecteuclid.org/euclid.acta/1485891049

Digital Object Identifier
doi:10.1007/BF02392693

Zentralblatt MATH identifier
0897.34074

Rights
1997 © Institut Mittag-Leffler

Citation

Gordon, A. Y.; Jitomirskaya, S.; Last, Y.; Simon, B. Duality and singular continuous spectrum in the almost Mathieu equation. Acta Math. 178 (1997), no. 2, 169--183. doi:10.1007/BF02392693. http://projecteuclid.org/euclid.acta/1485891049.


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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, in Solitons 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, in XIth 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, in XIth 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, in Mathematical 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.