Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The Schrödinger equation with a quasi-periodic potential

Author: Steve Surace
Journal: Trans. Amer. Math. Soc. 320 (1990), 321-370
MSC: Primary 34B25; Secondary 47E05, 81C05, 82A42
MathSciNet review: 998358
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the Schràdinger equation

$\displaystyle - \frac{{{d^2}}} {{d{x^2}}}\psi + \varepsilon (\cos x + \cos (\alpha x + \vartheta ))\psi = E\psi $

where $ \varepsilon$ is small and $ \sigma$ satisfies the Diophantine inequality

$\displaystyle \vert p + q\alpha \vert \geq C/{q^2}{\text{for}}p{\text{,}}q \in {\mathbf{Z}},q \ne 0.$

. We look for solutions of the form

$\displaystyle \psi (x) = {e^{iKx}}q(x) = {e^{iKx}}\sum {{\psi _{mn}}{e^{inx}}} {e^{im(\alpha x + \vartheta )}}$

. If we try to solve for $ \psi = {\psi _{mn}}$ we are led to the Schràdinger equation on the lattice $ {{\mathbf{Z}}^2}$

$\displaystyle H(K)\psi = (\varepsilon \Delta + V(K))\psi = E\psi $

where $ \Delta$ is the discrete Laplacian (without diagonal terms) and $ V(K)$ is some potential on $ {{\mathbf{Z}}^2}$ . We have two main results:

(1) For $ \varepsilon$ sufficiently small, $ H(K)$ has pure point spectrum for almost every $ K$.

(2) For $ \varepsilon$ sufficiently small, the operator

$\displaystyle - {d^2}/d{x^2} + \varepsilon (\cos x + \cos (\alpha x + \vartheta ))$

has no point spectrum.

To prove our results, we must get decay estimates on the Green's function $ {(E - H)^{ - 1}}$. The decay of the eigenfunction follows from this. In general, we must keep track of small divisors which can make the Green's function large. This is accomplished by a KAM (Kolmogorov, Arnold, Moser) type of multiscale perturbation analysis.

References [Enhancements On Off] (What's this?)

  • [1] J. M. Berezanskii, Expansion in eigenfunctions of self adjoint operators, Transl. Math. Monographs, vol. 17, Amer. Math. Soc., Providence, R.I., 1968.
  • [2] F. Bloch, Über die Quantenmechanik der Elektronen in Kristallgittern, Z. Phys. 52 (1928), 555.
  • [3] F. Delyon, Y. Levy, and B. Souillard, Comm. Math. Phys. 100 (1985), 463. MR 806247 (86m:82042)
  • [4] E. I. Dinaburg and Ya. G. Sinai, The one-dimensional Schràdinger equation with a quasiperiodic potential, Functional Anal. Appl. 9 (1975), 279.
  • [5] J. Fràhlich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Comm. Math. Phys. 88 (1983), 151. MR 696803 (85c:82004)
  • [6] J. Fràhlich, F. Martinelli, E. Scoppola, and T. Spencer, Localization in the Anderson tight binding model, Comm. Math. Phys. 101 (1985), 21. MR 814541 (87a:82047)
  • [7] G. Gallavotti, The elements of mechanics, Springer-Verlag, New York, 1983. MR 698947 (85g:70001)
  • [8] I. Gold'sheid, S. Molchanov, and L. Pastur, Pure point spectrum of stochastic one-dimensional Schràdinger operators, Functional Anal. Appl. 11 (1977), 1.
  • [9] R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys. 84 (1982), 403. MR 667409 (83h:34018)
  • [10] T. Kato, Perturbation theory for linear operators, 2nd ed., Springer, New York, 1976. MR 0407617 (53:11389)
  • [11] J. Moser and J. Pàschel, An extension of a result by Dinaburg and Sinai on quasi-periodic potentials, Comment. Math. Helv. 59 (1984), 39. MR 743943 (85m:34064)
  • [12] B. Simon, Almost periodic Schràdinger operators: a review, Adv. in Appl. Math. 3 (1982), 463. MR 682631 (85d:34030)
  • [13] B. Simon and T. Wolff, Comm. Pure Appl. Math. 39 (1986), 75. MR 820340 (87k:47032)
  • [14] Ya. G. Sinai, Anderson localization for the one dimensional difference Schràdinger operator with quasiperiodic potential, J. Statist. Phys. 46 (1987), 861. MR 893122 (88h:82016)
  • [15] T. Spencer, The Schràdinger equation with a random potential: a mathematical review, Lecture Notes, Les Houches Summer School, 1984.
  • [16] T. Spencer, J. Fràhlich, and P. Wittwer, Localization for a class of one dimensional quasiperiodic Schràdinger operators, preprint, 1987.
  • [17] E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations, Oxford Univ. Press, 1962. MR 0176151 (31:426)
  • [18] G. André and S. Aubry, Analyticity breaking and Anderson localization in incommensurate lattices, Ann. Israel Phys. Soc. 3 (1980), 133. MR 626837 (83b:82076)
  • [19] S. Aubry, The new concept of transition by breaking of analyticity, Solid State Sci. 8 (1978), 264. MR 522493 (80d:82034)
  • [20] F. Delyon, Absence of localization in the almost Mathieu equation, J. Phys. A 20 (1987), L21. MR 873175 (88d:81009)
  • [21] P. Lax, Lecture notes on Hilbert space, New York Univ., 1970.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 34B25, 47E05, 81C05, 82A42

Retrieve articles in all journals with MSC: 34B25, 47E05, 81C05, 82A42

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society