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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Spectral analysis of the generalized surface Maryland model


Authors: F. Bentosela, Ph. Briet and L. Pastur
Original publication: Algebra i Analiz, tom 16 (2004), nomer 6.
Journal: St. Petersburg Math. J. 16 (2005), 923-942
MSC (2000): Primary 35J10, 35P25
Published electronically: November 17, 2005
MathSciNet review: 2117447
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Abstract: The $ d$-dimensional discrete Schrödinger operator whose potential is supported on the subspace $ \mathbb{Z}^{d_{2}}$ of $ \mathbb{Z}^{d} $ is considered: $ H=H_{a}+V_{M}$, where $ H_{a}=H_{0}+V_{a}$, $ H_{0}$ is the $ d$-dimensional discrete Laplacian, $ V_{a}$ is a constant ``surface'' potential, $ V_{a}(\mathbf{x})=a\delta (x_{1})$, $ \mathbf{x}=(x_{1},x_{2})$, $ x_{1}\in \mathbb{Z}^{d_{1}}$, $ x_{2}\in \mathbb{Z}^{d_{2}}$, $ d_{1}+d_{2}=d$, and $ V_{M}(\mathbf{x})=g\delta (x_{1})\tan \pi (\alpha \cdot x_{2}+\omega )$ with $ \alpha \in \mathbb{R}^{d_{2}}$, $ \omega \in \lbrack 0,1)$. It is proved that if the components of $ \alpha $ are rationally independent, i.e., the surface potential is quasiperiodic, then the spectrum of $ H$ on the interval $ [-d,d] $ (coinciding with the spectrum of the discrete Laplacian) is purely absolutely continuous, and the associated generalized eigenfunctions have the form of the sum of the incident wave and waves reflected by the surface potential and propagating into the bulk of $ \mathbb{Z}^{d}$. If, in addition, $ \alpha $ satisfies a certain Diophantine condition, then the remaining part $ \mathbb{R}\setminus \lbrack -d,d]$ of the spectrum is pure point, dense, and of multiplicity one, and the associated eigenfunctions decay exponentially in both $ x_{1}$ and $ x_{2}$ (localized surface states). Also, the case of a rational $ \alpha =p/q$ for $ d_{1}=d_{2}=1$ (i.e., the case of a periodic surface potential) is discussed. In this case the entire spectrum is purely absolutely continuous, and besides the bulk waves there are also surface waves whose amplitude decays exponentially as $ \vert x_{1}\vert\rightarrow \infty $ but does not decay in $ x_{2}$. The part of the spectrum corresponding to the surface states consists of $ q$ separated bands. For large $ q$, the bands outside of $ [-d,d]$ are exponentially small in $ q$, and converge in a natural sense to the pure point spectrum of the quasiperiodic case with Diophantine $ \alpha $'s.


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Additional Information

F. Bentosela
Affiliation: Centre de Physique Théorique, Luminy, Case 907, Marseille 13288, France
Email: Francois.Bentosela@cpt.univ-mrs.fr

Ph. Briet
Affiliation: U. F. R. de Mathématiques, Université Paris 7, 2, Pl. Jussieu, Paris 75251, France
Email: briet@cpt.univ-mrs.fr

L. Pastur
Affiliation: Institute for Low Temperature Physics, Kharkiv, Ukraine
Email: pastur@math.jussieu.fr

DOI: http://dx.doi.org/10.1090/S1061-0022-05-00884-8
PII: S 1061-0022(05)00884-8
Keywords: Discrete Schr\"odinger operator, Maryland model
Received by editor(s): March 17, 2004
Published electronically: November 17, 2005
Dedicated: Dedicated to M. S. Birman on the occasion of his 75th birthday
Article copyright: © Copyright 2005 American Mathematical Society