The rotation number for almost periodic potentials. (English) Zbl 0497.35026

Johnson, R.; Moser, Jürgen

Commun. Math. Phys. 84, 403-438 (1982); erratum ibid. 90, 317-318 (1983).

MSC:

35J10	Schrödinger operator, Schrödinger equation
34L99	Ordinary differential operators
81Q99	General mathematical topics and methods in quantum theory
35Q99	Partial differential equations of mathematical physics and other areas of application

Keywords:

density of states; almost periodic potential; rotation number; spectral resolution; conservation laws; Korteweg-de Vries equation

References:

[1]	Avron, J., Simon, B.: Cantor sets and Schrödinger operators I. Transient and recurrent spectrum. Preprint 1980
[2]	Avron, J., Simon, B.: Cantor sets and Schrödinger operators II. The density of states and the Andre-Aubrey theorem. (in preparation)
[3]	Krylov, N., Bogoliuboff, N.: La théorie générale de la mesure et sont application à l’étude des systèmes dynamiques de la méchanique non linéaire. Ann. Math.38 65-113 (1937) · Zbl 0016.08604
[4]	Bohr, H.: Fastperiodische Funktionen. Erg. Math.1 5 (1932) · JFM 58.0264.01
[5]	Bourbaki, N.: Integration, Vol. V, Paris: Hermann 1965
[6]	Coddington, E., Levinson, N.: Theory of ordinary differential equations. New York: McGraw-Hill 1955 · Zbl 0064.33002
[7]	Dubrovin, B., Matveev, V., Novikov, S.: Nonlinear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian varieties, Russ. Math. Surv.31 59-146 (1976) · Zbl 0346.35025
[8]	Fink, A.: Almost periodic differential equations. Lecture Notes in Mathematics.377, Berlin, Heidelberg, New York: Springer 1974 · Zbl 0325.34039
[9]	Gordon, A.: On the point spectrum of the one-dimensional Schrödinger operator. Russ. Math. Surv.31 257-258 (1976) · Zbl 0342.34012
[10]	Hille, E.: Lectures on ordinary differential equations. Reading, Mass: Addison-Wesley 1969 · Zbl 0179.40301
[11]	Pastur, L.: Spectrum of random selfadjoint operators. Usp. Math. Nauk28 (1973) 3-64, or Russ. Math. Surv.28, 1-67 (1973)
[12]	Johnson, R.: The recurrent Hill’s equation. J. Diff. Equations (to appear) · Zbl 0535.34021
[13]	Lax, P.: Almost periodic solutions of the KdV equation. SIAM Rev.18 351-375 (1976) · Zbl 0329.35015
[14]	McKean, H., van Moerbeke, P.: The spectrum of Hill’s equation. Inv. Math.30 217-274 (1975) · Zbl 0319.34024
[15]	McKean, H., Trubowitz, E.: Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points. Commun. Pure Appl. Math.29 153-226 (1976) · Zbl 0339.34024
[16]	Millonshchikov, V.: Proof of the existence of irregular systems of linear differential equations with almost periodic coefficients. Diff. Equations4 203-205 (1968)
[17]	Moser, J.: An example of a Schrödinger operator with almost periodic potential and nowhere dense spectrum, Comm. Math. Helv.56 198-224 (1981) · Zbl 0477.34018
[18]	Nemytskii, V., Stepanov, V.: Qualitative theory of differential equations. Princeton: Princeton Univ. Press 1960 · Zbl 0089.29502
[19]	Pastur, L.: Spectral properties of disordered systems in the one-body approximation. Commun. Math. Phys.75 179-196 (1980) · Zbl 0429.60099
[20]	Sacker, R., Sell, G.: Dichotomies and invariant splittings for linear differential systems I, J. Diff. Equations15 429-458 (1974) · Zbl 0294.58008
[21]	Sacker, R., Sell, G.: A spectral theory for linear differential systems. J. Diff. Equations27 320-358 (1978) · Zbl 0372.34027
[22]	Sarnak, P.: Spectral behaviour of quasi-periodic potentials, Commun. Math. Phys. (to appear) · Zbl 0506.35074
[23]	Scharf, G.: Fastperiodische Potentiale. Helv. Phys. Acta24 573-605 (1965) · Zbl 0145.10801
[24]	Selgrade, J.: Isolated invariant sets for lows on vector bundles. Trans. Am. Math. Soc. 359-390 (1975) · Zbl 0265.58004
[25]	Schwartzman, S.: Asymptotic Cycles. Ann. Math.66 270-284 (1957) · Zbl 0207.22603