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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.

References [Enhancements On Off] (What's this?)

  • 1. N. I. Akhiezer and I. M. Glazman, The theory of linear operators in Hilbert space, GITTL, Moscow-Leningrad, 1950; English transl., Vol. 2, F. Ungar Publ. Co., New York, 1963. MR 0044034 (13:3586); MR 0264421 (41:9015b)
  • 2. F. Bentosela, Ph. Briet, and L. Pastur, On the spectral and wave propagation properties of the surface Maryland model, J. Math. Phys. 44 (2003), 1-35. MR 1946689 (2003k:81049)
  • 3. A. Boutet de Monvel and A. Surkova, Localisation des états de surface pour une classe d'opérateurs de Schrödinger discrets à potentiels de surface quasi-périodiques, Helv. Phys. Acta 71 (1998), 459-490. MR 1651042 (2000a:47070a)
  • 4. I. P. Kornfel'd, Ya. G. Sinai, and S. V. Fomin, Ergodic theory, ``Nauka'', Moscow, 1980; English transl., Grundlehren Math. Wiss., vol. 245, Springer-Verlag, New York, 1982. MR 0610981 (83a:28017); MR 0832433 (87f:28019)
  • 5. E. B. Davies and B. Simon, Scattering theory for systems with different spatial asymptotics on the left and right, Comm. Math. Phys. 63 (1978), 277-301. MR 0513906 (80c:81110)
  • 6. A. L. Figotin and L. A. Pastur, An exactly solvable model of a multidimensional incommensurate structure, Comm. Math. Phys. 95 (1984), 401-425. MR 0767188 (86f:81027)
  • 7. -, Spectra of random and almost-periodic operators, Grundlehren Math. Wiss., vol. 297, Springer-Verlag, Berlin, 1992. MR 1223779 (94h:47068)
  • 8. V. Grinshpun, Localization for random potentials supported on a subspace, Lett. Math. Phys. 34 (1995), 103-117. MR 1335579 (96e:82052)
  • 9. A. Grossmann, R. Høegh-Krohn, and M. Mebkhout, The one particle theory of periodic point interactions. Polymers, monomolecular layers, and crystals, Comm. Math. Phys. 77 (1980), 87-110. MR 0588688 (82i:81022)
  • 10. V. Jakšic and Y. Last, Corrugated surfaces and a.c. spectrum, Rev. Math. Phys. 12 (2000), 1465-1503. MR 1809458 (2001m:47143)
  • 11. -, Surface states and spectra, Comm. Math. Phys. 218 (2001), 459-477. MR 1828849 (2002g:81030)
  • 12. V. Jakšic and S. Molchanov, On the spectrum of the surface Maryland model, Lett. Math. Phys. 45 (1998), 189-193. MR 1641176 (99h:82046)
  • 13. -, On the surface spectrum in dimension two, Helv. Phys. Acta 71 (1998), 629-657. MR 1669046 (2000b:81031)
  • 14. -, Localization of surface spectra, Comm. Math. Phys. 208 (1999), 153-172. MR 1729882 (2000k:47048)
  • 15. -, Wave operators for the surface Maryland model, J. Math. Phys. 41 (2000), 4452-4463. MR 1765613 (2002a:81052)
  • 16. V. Jakšic, S. Molchanov, and L. Pastur, On the propagation properties of surface waves, Wave Propagation in Complex Media (Minneapolis, MN, 1994), IMA Vol. Math. Appl., vol. 96, Springer, New York, 1998, pp. 143-154. MR 1489748
  • 17. Yu. E. Karpeshina, The spectrum and eigenfunctions of the Schrödinger operator in a three-dimensional space with point-like potential of the homogeneous two-dimensional lattice type, Teoret. Mat. Fiz. 57 (1983), no. 3, 414-423; English transl., Theoret. and Math. Phys. 57 (1983), no. 3, 1231-1238. MR 0735399 (85f:81011)
  • 18. -, An eigenfunction expansion theorem for the Schrödinger operator with a homogeneous simple two-dimensional lattice of potentials of zero radius in a three-dimensional space, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1984, vyp. 1, 11-17; English transl. in Vestnik Leningrad Univ. Math. 17. MR 0743579 (85h:35158)
  • 19. T. Kato, Perturbation theory for linear operators, Grundlehren Math. Wiss., vol. 132, Springer-Verlag, New York, 1966. MR 0203473 (34:3324)
  • 20. B. Khoruzhenko and L. Pastur, Localization of surface states: an explicitly solvable model, Phys. Rep. 288 (1997), 109-125.
  • 21. B. Simon, Almost periodic Schrödinger operators. IV. The Maryland model, Ann. Physics 159 (1985), 157-183. MR 0776654 (86m:81038)
  • 22. D. Yafaev, Mathematical scattering theory. General theory, S.-Peterburg. Gos. Univ., St. Petersburg, 1994; English transl., Amer. Math. Soc., Providence, RI, 1999. MR 1784870 (2001e:47015)

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Additional Information

F. Bentosela
Affiliation: Centre de Physique Théorique, Luminy, Case 907, Marseille 13288, France

Ph. Briet
Affiliation: U. F. R. de Mathématiques, Université Paris 7, 2, Pl. Jussieu, Paris 75251, France

L. Pastur
Affiliation: Institute for Low Temperature Physics, Kharkiv, Ukraine

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

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