Spectral analysis of the generalized surface Maryland model
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- by F. Bentosela, Ph. Briet and L. Pastur
- St. Petersburg Math. J. 16 (2005), 923-942
- DOI: https://doi.org/10.1090/S1061-0022-05-00884-8
- Published electronically: November 17, 2005
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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 $|x_{1}|\rightarrow \infty$ 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
- N. I. Ahiezer and I. M. Glazman, Teoriya lineĭnyh operatorov v gil′bertovom prostranstve, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950 (Russian). MR 0044034
- F. Bentosela, Ph. Briet, and L. Pastur, On the spectral and wave propagation properties of the surface Maryland model, J. Math. Phys. 44 (2003), no. 1, 1–35. MR 1946689, DOI 10.1063/1.1521798
- Anne Boutet de Monvel and Anna 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), no. 5, 459–490 (French, with English summary). MR 1651042
- I. P. Kornfel′d, Ya. G. Sinaĭ, and S. V. Fomin, Ergodicheskaya teoriya, “Nauka”, Moscow, 1980 (Russian). MR 610981
- E. B. Davies and B. Simon, Scattering theory for systems with different spatial asymptotics on the left and right, Comm. Math. Phys. 63 (1978), no. 3, 277–301. MR 513906
- A. L. Figotin and L. A. Pastur, An exactly solvable model of a multidimensional incommensurate structure, Comm. Math. Phys. 95 (1984), no. 4, 401–425. MR 767188
- Leonid Pastur and Alexander Figotin, Spectra of random and almost-periodic operators, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 297, Springer-Verlag, Berlin, 1992. MR 1223779, DOI 10.1007/978-3-642-74346-7
- V. Grinshpun, Localization for random potentials supported on a subspace, Lett. Math. Phys. 34 (1995), no. 2, 103–117. MR 1335579, DOI 10.1007/BF00739090
- 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), no. 1, 87–110. MR 588688
- Vojkan Jakšić and Yoram Last, Corrugated surfaces and a.c. spectrum, Rev. Math. Phys. 12 (2000), no. 11, 1465–1503. MR 1809458, DOI 10.1142/S0129055X00000563
- Vojkan Jak ić and Yoram Last, Surface states and spectra, Comm. Math. Phys. 218 (2001), no. 3, 459–477. MR 1828849, DOI 10.1007/PL00005560
- Vojkan Jakšić and Stanislav Molchanov, On the spectrum of the surface Maryland model, Lett. Math. Phys. 45 (1998), no. 3, 189–193. MR 1641176, DOI 10.1023/A:1007579806383
- Vojkan Jakšić and Stanislav Molchanov, On the surface spectrum in dimension two, Helv. Phys. Acta 71 (1998), no. 6, 629–657. MR 1669046
- Vojkan Jakšić and Stanislav Molchanov, Localization of surface spectra, Comm. Math. Phys. 208 (1999), no. 1, 153–172. MR 1729882, DOI 10.1007/s002200050752
- Vojkan Jakšić and Stanislav Molchanov, Wave operators for the surface Maryland model, J. Math. Phys. 41 (2000), no. 7, 4452–4463. MR 1765613, DOI 10.1063/1.533353
- V. Jakšić, 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, DOI 10.1007/978-1-4612-1678-0_{7}
- 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 (Russian, with English summary). MR 735399
- Yu. E. Karpeshina, 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. vyp. 1 (1984), 11–17 (Russian, with English summary). MR 743579
- Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
- B. Khoruzhenko and L. Pastur, Localization of surface states: an explicitly solvable model, Phys. Rep. 288 (1997), 109–125.
- Barry Simon, Almost periodic Schrödinger operators. IV. The Maryland model, Ann. Physics 159 (1985), no. 1, 157–183. MR 776654, DOI 10.1016/0003-4916(85)90196-4
- D. R. Yafaev, Matematicheskaya teoriya rasseyaniya, Izdatel′stvo Sankt-Peterburgskogo Universiteta, St. Petersburg, 1994 (Russian, with Russian summary). Obshchaya teoriya. [General theory]. MR 1784870
Bibliographic 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
- Received by editor(s): March 17, 2004
- Published electronically: November 17, 2005
- © Copyright 2005 American Mathematical Society
- Journal: St. Petersburg Math. J. 16 (2005), 923-942
- MSC (2000): Primary 35J10, 35P25
- DOI: https://doi.org/10.1090/S1061-0022-05-00884-8
- MathSciNet review: 2117447
Dedicated: Dedicated to M. S. Birman on the occasion of his 75th birthday