×

zbMATH — the first resource for mathematics

Almost periodic Schrödinger operators. III: The absolutely continuous spectrum in one dimension. (English) Zbl 0562.35026
Summary: [For parts I and II see J. Avron and the second author, ibid. 82, 101-120 (1981; Zbl 0484.35069) and Duke Math. J. 50, 369-391 (1983; Zbl 0544.35030) respectively.]
We discuss the absolutely continuous spectrum of \(H=-d^ 2/dx^ 2+V(x)\) with V almost periodic and its discrete analog \((hu)(n)=u(n+1)+u(n- 1)+V(n)u(n).\) Especial attention is paid to the set, A, of energies where the Lyaponov exponent vanishes. This set is known to be the essential support of the a.c. part of the spectral measure.
We prove for a.e. V in the hull and a.e. E in A, H and h have continuum eigenfunctions, u, with \(| u|\) almost periodic. In the discrete case, we prove that \(| A| \leq 4\) with equality only if \(V=const.\) If k is the integrated density of states, we prove that on A, \(2kdk/dE\geq \pi^{-2}\) in the continuum case and that \(2\pi \sin \pi kdk/dE\geq 1\) in the discrete case. We also provide a new proof of the Pastur-Ishii theorem and that the multiplicity of the absolutely continuous spectrum is 2.

MSC:
35J10 Schrödinger operator, Schrödinger equation
35P05 General topics in linear spectral theory for PDEs
35B15 Almost and pseudo-almost periodic solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Avron, J., Simon, B.: Transient and recurrent spectrum. J. Funct. Anal.43, 1-31 (1981) · Zbl 0488.47021 · doi:10.1016/0022-1236(81)90034-3
[2] Avron, J., Simon, B.: Almost periodic Schrödinger operators. II. The density of states. Duke Math. J.50, 369-391 (1983) · Zbl 0544.35030 · doi:10.1215/S0012-7094-83-05016-0
[3] Davies, E.B., Simon, B.: Scattering theory for systems with different spatial asymptotics on the left and right. Commun. Math. Phys.63, 277-301 (1978) · Zbl 0393.34015 · doi:10.1007/BF01196937
[4] Dinaburg, E.I., Sinai, Ya.G.: On the one dimensional Schrödinger equcation with quasiperiodic potential. Funkt. Anal. i Priloz.9, 8-21 (1975) · Zbl 0357.58011 · doi:10.1007/BF01078168
[5] Gordon, A. Ya.: On the point spectrum of the one-dimensional Schrödinger operator. Usp. Math. Nauk.31, 257 (1976)
[6] Herbert, D., Jones, R.: Localized states in disordered systems. J. Phys. C4, 1145-1161 (1971)
[7] Ishii, K.: Localization of eigenstates and transport phenomena in the one dimensional disordered system. Supp. Theor. Phys.53, 77-138 (1973) · doi:10.1143/PTPS.53.77
[8] Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys.84, 403-438 (1982) · Zbl 0497.35026 · doi:10.1007/BF01208484
[9] Kirsch, W., Martinelli, F.: On the spectrum of Schrödinger operators with a random potential. Commun. Math. Phys.85, 329 (1982) · Zbl 0506.60058 · doi:10.1007/BF01208718
[10] Kotani, S.: Lyaponov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators. Proc. Kyoto Stoch. Conf., 1982
[11] Kunz, H., Souillard, B.: On the spectrum of random finite difference operators. Commun. Math. Phys.76, 201-246 (1980) · Zbl 0449.60048 · doi:10.1007/BF01942371
[12] Moser, J.: An example of a Schrödinger operator with almost periodic potential and nowhere dense spectrum. Commun. Math. Helv.56, 198-224 (1981) · Zbl 0477.34018 · doi:10.1007/BF02566210
[13] Pastur, L.: Spectral properties of disordered systems in the one body approximation. Commun. Math. Phys.75, 179-196 (1980) · Zbl 0429.60099 · doi:10.1007/BF01222516
[14] Reed, M., Simon, B.: Methods in modern mathematical physics, Vol. III: Scattering theory. New York: Academic Press 1978 · Zbl 0401.47001
[15] Saks, J.: Theory of the integral. New York: G.E. Strechert Co. 1937 · Zbl 0017.30004
[16] Simon, B.: Schrödinger semigroups. Bull. Am. Math. Soc.7, 447-526 (1982) · Zbl 0524.35002 · doi:10.1090/S0273-0979-1982-15041-8
[17] Simon, B.: Almost periodic Schrödinger operators: a review. Adv. Appl. Math.3, 463-490 (1982) · Zbl 0545.34023 · doi:10.1016/S0196-8858(82)80018-3
[18] Simon, B.: Kotani theory for one dimensional stochastic Jacobi matrices. Commun. Math. Phys.89, 227-234 (1983) · Zbl 0534.60057 · doi:10.1007/BF01211829
[19] Thouless, D.: A relation between the density of states and range of localization for one-dimensional random systems. J. Phys. C5, 77-81 (1972)
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.