Absolutely continuous spectrum and scattering in the surface Maryland model

Bentosela, François; Briet, Philippe; Pastur, Leonid

Journées équations aux dérivées partielles (2001), article no. 10, 15 p.

We study the discrete Schrödinger operator $H$ in ${𝐙}^d$ with the surface quasi periodic potential $V\left(x\right)=g\delta \left(x_1\right)tan\pi (\alpha ·x_2+\omega )$, where $x=(x_1,x_2),x_1\in {𝐙}^{d_1},x_2\in {𝐙}^{d_2},\alpha \in {𝐑}^{d_2},\omega \in [0,1)$. We first discuss a proof of the pure absolute continuity of the spectrum of $H$ on the interval $[-d,d]$ (the spectrum of the discrete laplacian) in the case where the components of $\alpha$ are rationally independent. Then we show that in this case the generalized eigenfunctions have the form of the “volume” waves, i.e. of the sum of the incident plane wave and reflected from the hyper-plane ${𝐙}^{d_1}$ waves, the form that is well known in the scattering theory for decaying potential. These eigenfunctions are orthogonal, complete and verify a natural analogue of the Lippmann-Schwinger equation. We find the wave operators and the scattering matrix in this case. We discuss also the case of rational $\alpha =p/q$’s, $p,q\in \mathbb{N}$ for $d_1=d_2=1$, i.e. of a periodic surface potential. In this case besides the volume waves there are also the surface waves, whose amplitude decays exponentially as $|x_1|\to \infty$. For large $q$ corresponding part of the absolutely continuos spectrum consists of $q$ exponentially narrow bands, lying all except one outside the interval $[-2,2]$, and converging in a natural sense as $q\to \infty$ to the dense point spectrum found before in [13] for the irrational diophantine $\alpha$’s.

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