### Abstract

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|≦4 with equality only if V=const. If k is the integrated density of states, we prove that on A, 2 kdk/dE≧π^{-2} in the continuum case and that 2π sinπkdk/dE≧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.

Original language | English (US) |
---|---|

Pages (from-to) | 389-411 |

Number of pages | 23 |

Journal | Communications in Mathematical Physics |

Volume | 90 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1983 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

**Almost periodic Schrödinger operators - III. The absolutely continuous spectrum in one dimension.** / Deift, P.; Simon, B.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 90, no. 3, pp. 389-411. https://doi.org/10.1007/BF01206889

}

TY - JOUR

T1 - Almost periodic Schrödinger operators - III. The absolutely continuous spectrum in one dimension

AU - Deift, P.

AU - Simon, B.

PY - 1983/9

Y1 - 1983/9

N2 - We discuss the absolutely continuous spectrum of H=-d2/dx2+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|≦4 with equality only if V=const. If k is the integrated density of states, we prove that on A, 2 kdk/dE≧π-2 in the continuum case and that 2π sinπkdk/dE≧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.

AB - We discuss the absolutely continuous spectrum of H=-d2/dx2+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|≦4 with equality only if V=const. If k is the integrated density of states, we prove that on A, 2 kdk/dE≧π-2 in the continuum case and that 2π sinπkdk/dE≧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.

UR - http://www.scopus.com/inward/record.url?scp=0001091673&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001091673&partnerID=8YFLogxK

U2 - 10.1007/BF01206889

DO - 10.1007/BF01206889

M3 - Article

VL - 90

SP - 389

EP - 411

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -