# Absolutely continuous spectrum and scattering in the surface Maryland model

François Bentosela; Philippe Briet; Leonid Pastur

Journées équations aux dérivées partielles (2001)

- page 1-15
- ISSN: 0752-0360

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topBentosela, François, Briet, Philippe, and Pastur, Leonid. "Absolutely continuous spectrum and scattering in the surface Maryland model." Journées équations aux dérivées partielles (2001): 1-15. <http://eudml.org/doc/93407>.

@article{Bentosela2001,

abstract = {We study the discrete Schrödinger operator $H$ in $\mathbf \{Z\}^d$ with the surface quasi periodic potential $V(x)=g \delta (x_1) \tan \pi (\alpha \cdot x_2+ \omega )$, where $x=(x_1,x_2), \ x_1 \in \mathbf \{Z\}^\{d_1\}, \ x_2 \in \mathbf \{Z\}^\{d_2\}, \ \alpha \in \mathbf \{R\}^\{d_2\}, \ \omega \in [0,1)$. 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 $\mathbf \{Z\}^\{d_1\}$ 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| \rightarrow \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 \rightarrow \infty $ to the dense point spectrum found before in [13] for the irrational diophantine $\alpha $’s.},

author = {Bentosela, François, Briet, Philippe, Pastur, Leonid},

journal = {Journées équations aux dérivées partielles},

keywords = {discrete Schrödinger operator; quasi-periodic potential; absolute continuity; spectrum; completeness; wave operator; surface waves},

language = {eng},

pages = {1-15},

publisher = {Université de Nantes},

title = {Absolutely continuous spectrum and scattering in the surface Maryland model},

url = {http://eudml.org/doc/93407},

year = {2001},

}

TY - JOUR

AU - Bentosela, François

AU - Briet, Philippe

AU - Pastur, Leonid

TI - Absolutely continuous spectrum and scattering in the surface Maryland model

JO - Journées équations aux dérivées partielles

PY - 2001

PB - Université de Nantes

SP - 1

EP - 15

AB - We study the discrete Schrödinger operator $H$ in $\mathbf {Z}^d$ with the surface quasi periodic potential $V(x)=g \delta (x_1) \tan \pi (\alpha \cdot x_2+ \omega )$, where $x=(x_1,x_2), \ x_1 \in \mathbf {Z}^{d_1}, \ x_2 \in \mathbf {Z}^{d_2}, \ \alpha \in \mathbf {R}^{d_2}, \ \omega \in [0,1)$. 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 $\mathbf {Z}^{d_1}$ 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| \rightarrow \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 \rightarrow \infty $ to the dense point spectrum found before in [13] for the irrational diophantine $\alpha $’s.

LA - eng

KW - discrete Schrödinger operator; quasi-periodic potential; absolute continuity; spectrum; completeness; wave operator; surface waves

UR - http://eudml.org/doc/93407

ER -

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