Absolutely continuous spectrum and scattering in the surface Maryland model
Journées équations aux dérivées partielles (2001), article no. 10, 15 p.

We study the discrete Schrödinger operator H in 𝐙 d with the surface quasi periodic potential V(x)=gδ(x 1 )tanπ(α·x 2 +ω), where x=(x 1 ,x 2 ),x 1 𝐙 d 1 ,x 2 𝐙 d 2 ,α𝐑 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 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.

     author = {Bentosela, Fran\c{c}ois and Briet, Philippe and Pastur, Leonid},
     title = {Absolutely continuous spectrum and scattering in the surface {Maryland} model},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {10},
     publisher = {Universit\'e de Nantes},
     year = {2001},
     doi = {10.5802/jedp.594},
     zbl = {1026.47024},
     mrnumber = {1843411},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jedp.594/}
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
DA  - 2001///
PB  - Université de Nantes
UR  - http://archive.numdam.org/articles/10.5802/jedp.594/
UR  - https://zbmath.org/?q=an%3A1026.47024
UR  - https://www.ams.org/mathscinet-getitem?mr=1843411
UR  - https://doi.org/10.5802/jedp.594
DO  - 10.5802/jedp.594
LA  - en
ID  - JEDP_2001____A10_0
ER  - 
%0 Journal Article
%A Bentosela, François
%A Briet, Philippe
%A Pastur, Leonid
%T Absolutely continuous spectrum and scattering in the surface Maryland model
%J Journées équations aux dérivées partielles
%D 2001
%I Université de Nantes
%U https://doi.org/10.5802/jedp.594
%R 10.5802/jedp.594
%G en
%F JEDP_2001____A10_0
Bentosela, François; Briet, Philippe; Pastur, Leonid. Absolutely continuous spectrum and scattering in the surface Maryland model. Journées équations aux dérivées partielles (2001), article  no. 10, 15 p. doi : 10.5802/jedp.594. http://archive.numdam.org/articles/10.5802/jedp.594/

[FKS] Cornfeld, I. P., Fomin, S. V., Sinai, Ya. G. Ergodic Theory. Springer-Verlag, New York, 1982 | MR | Zbl

[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 | MR | Zbl

[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. | MR

[JL1] Jakšić, V., Last, Y. Spectral structure of Anderson type Hamiltonians Invent. Math. 141 (2000) 561-577 | MR | Zbl

[JL2] Jakšić, V., Last, Y. Corrugated surfaces and a.c. spectrum. Rev. Math. Phys. 12 (2000) 1465-1503 | MR | Zbl

[JM1] Jakšić, V., Molchanov, S. On the spectrum of the surface Maryland model. Lett. Math. Phys. 45 1998 189-193 | MR | Zbl

[JM2] Jakšić, V., Molchanov, S. On the surface spectrum in dimension two. Helv. Phys. Acta 71 1998 629-657 | MR | Zbl

[JM3] Jakšić, V., Molchanov, S. Localization of surface spectra. Comm. Math. Phys. 208 1999 153-172 | MR | Zbl

[JM4] Jakšić, V., Molchanov, S. Wave operators for the surface Maryland model. J. Math. Phys. 41 (2000) 4452-4463 | MR | Zbl

[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 | MR | Zbl

[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 | MR

[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. | MR | Zbl

[KP] Khoruzhenko, B., Pastur, L. Localisation of surface states: an explicitly solvable model Physics Reports 288 1997 109-125. | Zbl

[Pa] Pastur, L Surface waves: propagation and localisation1995), Exp. No. VI, 12 pp. | Numdam | MR | Zbl

[PF] Pastur, L., Figotin A. Spectra of Random and Almost Periodic Operators Spriger Verlag, Berlin-Heidelberg, 1992 | MR | Zbl

[Pe] Pearson, D. B. Quantum Scattering and Spectral Theory. Academic Press, London, 1988 | MR | Zbl

[RS] Reed, M., Simon, B. Methods of Modern Mathematical Physics. III. Scattering Theory. Academic Press, New York-London, 1979 | MR | Zbl

[Si] Simon, B. Quantum Mechanics for Hamiltonians Defined as Quadratic Forms. Princeton University Press, Princeton, N. J., 1971. xv+244 pp. | MR | Zbl

[Si:84] Simon, B. Almost periodic Schrödinger operators. IV. The Maryland model. Ann. Physics 159 1985 157-183 | MR | Zbl

Cited by Sources: