## Abstract

We study spectral and scattering properties of the discrete Laplacian H on the half-space Z^{d+1}_{+} = Z^{d} × Z_{+} with boundary condition ψ(n, -1) = V(n)ψ(n, 0). We consider cases where V is a deterministic function and a random process on Z^{d}. Let H_{o} be the Dirichlet Laplacian on l^{2}(Z^{d+1}_{+}). We show that the wave operators Ω^{±}(H, Ho) exist for all V, and in particular, that σ(H_{o}) ⊂ σ_{ac}(H). We study when and where the wave operators are complete and the spectrum of H is purely absolutely continuous, and prove some optimal results. In particular, if V is a random process on a probability space (Ω, F, P), such that the random variables V(n) are independent and have densities, we show that the spectrum of H on σ(H_{0}) is purely absolutely continuous P-a.s. If in addition lim|V(n)| = ∞ P-a.s., we show that the wave operators Ω^{±}(H, H_{0}) are complete on σ(H_{0}) P-a.s.

Original language | English |
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Pages (from-to) | 1465-1503 |

Number of pages | 39 |

Journal | Reviews in Mathematical Physics |

Volume | 12 |

Issue number | 11 |

DOIs | |

State | Published - Nov 2000 |

### Bibliographical note

Funding Information:We are grateful to J. Derezinski, S. Molchanov, L. Pastur, J. Sahbani and B. Simon for useful discussions. In particular, we are grateful to B. Simon for convincing us at an early stage that Theorem 1.10 must hold, and to J. Sahbani for indicating to us some simpli cations of our original arguments. This work was partially supported by NATO Collaborative Research Grant CRG 970051. VJ’s work was also partially supported by NSERC. YL’s work was also partially supported by NSF grant DMS-9801474 and by an Allon fellowship.