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

Abstract

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We study the discrete Schrödinger operator H in 𝐙 d with the surface quasi periodic potential V ( x ) = g δ ( x 1 ) tan π ( α · x 2 + ω ) , where x = ( x 1 , x 2 ) , x 1 ∈ 𝐙 d 1 , x 2 ∈ 𝐙 d 2 , α ∈ 𝐑 d 2 , ω ∈ [ 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 α 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 α = p / q ’s, p , q ∈ 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 | → ∞ . 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 the dense point spectrum found before in [13] for the irrational diophantine α ’s.

How to cite

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Bentosela, 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 -

References

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  1. [FKS] Cornfeld, I. P., Fomin, S. V., Sinai, Ya. G.Ergodic Theory. Springer-Verlag, New York, 1982 Zbl0493.28007MR832433
  2. [PF:84] Figotin, A. L.; Pastur, L. A.An exactly solvable model of a multidimensional incommensurate structure. Comm. Math. Phys. 95 1984, no. 4, 401-425 Zbl0582.35101MR767188
  3. [Gr] Grossmann, A., Hoegh-Krohn, R., Mebkhout, M.The one particle theory of periodic point interactions. Polymers, monomolecular layers, and crystals Comm. Math. Phys. 77 1980 87-110. MR588688
  4. [JL1] Jakšić, V., Last, Y.Spectral structure of Anderson type Hamiltonians Invent. Math. 141 ( 2000) 561-577 Zbl0962.60056MR1779620
  5. [JL2] Jakšić, V., Last, Y.Corrugated surfaces and a.c. spectrum. Rev. Math. Phys. 12 ( 2000) 1465-1503 Zbl0979.60050MR1809458
  6. [JM1] Jakšić, V., Molchanov, S.On the spectrum of the surface Maryland model. Lett. Math. Phys. 45 1998 189-193 Zbl0916.47027MR1641176
  7. [JM2] Jakšić, V., Molchanov, S.On the surface spectrum in dimension two. Helv. Phys. Acta 71 1998 629-657 Zbl0939.60074MR1669046
  8. [JM3] Jakšić, V., Molchanov, S.Localization of surface spectra. Comm. Math. Phys. 208 1999 153-172 Zbl0952.60059MR1729882
  9. [JM4] Jakšić, V., Molchanov, S.Wave operators for the surface Maryland model. J. Math. Phys. 41 ( 2000) 4452-4463 Zbl0983.82007MR1765613
  10. [JMP] Jaksic, V., Molchanov, S., Pastur, L.On the propagation properties of surface waves In: IMA Vol. Math. Appl.96, Springer, New York, 1998, pp. 145-154 Zbl0905.65107MR1489748
  11. [Ka1] Karpeshina, Yu. E.The spectrum and eigenfunctions of the Schrödinger operator in a three-dimensional space with point-like potential of the homogeneous two-dimensional lattice type. (Russian) Teoret. Mat. Fiz. 57 1983 414-423 MR735399
  12. [Ka2] Karpeshina, Yu. E.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. (Russian) Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1984, vyp. 1, 11-17. Zbl0546.35020MR743579
  13. [KP] Khoruzhenko, B., Pastur, L.Localisation of surface states: an explicitly solvable model Physics Reports 288 1997 109-125. Zbl1001.82526
  14. [Pa] Pastur, LSurface waves: propagation and localisationIn: Journées “Equations aux Dérivées Partielles”, (Saint-Jean-de-Monts,1995), Exp. No. VI, 12 pp. Zbl0949.35512MR1360475
  15. [PF] Pastur, L., Figotin A.Spectra of Random and Almost Periodic Operators Spriger Verlag, Berlin-Heidelberg, 1992 Zbl0752.47002MR1223779
  16. [Pe] Pearson, D. B.Quantum Scattering and Spectral Theory. Academic Press, London, 1988 Zbl0673.47011MR1099604
  17. [RS] Reed, M., Simon, B.Methods of Modern Mathematical Physics. III. Scattering Theory. Academic Press, New York-London, 1979 Zbl0405.47007MR529429
  18. [Si] Simon, B.Quantum Mechanics for Hamiltonians Defined as Quadratic Forms. Princeton University Press, Princeton, N. J., 1971. xv+244 pp. Zbl0232.47053MR455975
  19. [Si:84] Simon, B.Almost periodic Schrödinger operators. IV. The Maryland model. Ann. Physics 159 1985 157-183 Zbl0595.35032MR776654

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