The spectra of the surface Maryland model for all phases
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- by Wencai Liu PDF
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Abstract:
We study the discrete Schrödinger operators $H_{\lambda ,\alpha ,\theta }$ on $\ell ^2(\mathbb {Z}^{d+1})$ with surface potential of the form $V(n,x)=\lambda \delta (x)\tan \pi (\alpha \cdot n+\theta )$, and $H_{\lambda ,\alpha ,\theta }^{+}$ on $\ell ^2(\mathbb {Z}^{d}\times \mathbb {Z}_+)$ with the boundary condition $\psi _{(n,-1)}=\lambda \tan \pi (\alpha \cdot n+\theta )\psi _{(n,0)}$, where $\alpha \in \mathbb {R}^d$ is rationally independent. We show that the spectra of $H_{\lambda ,\alpha ,\theta }$ and $H_{\lambda ,\alpha ,\theta }^{+}$ are $(-\infty ,\infty )$ for all parameters. We can also determine the absolutely continuous spectra and Hausdorff dimension of the spectral measures if $d=1$.References
- 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
- F. Bentosela, Ph. Briet, and L. Pastur, Spectral analysis of the generalized surface Maryland model, Algebra i Analiz 16 (2004), no. 6, 1–27; English transl., St. Petersburg Math. J. 16 (2005), no. 6, 923–942. MR 2117447, DOI 10.1090/S1061-0022-05-00884-8
- Ju. M. Berezans′kiĭ, Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968. Translated from the Russian by R. Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. MR 0222718
- H. Cycon, R. Froese, W. Kirsch, and B. Simon, Schrödinger operators with applications to quantum mechanics and global geometry, 1987.
- 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
- Shmuel Fishman, D. R. Grempel, and R. E. Prange, Chaos, quantum recurrences, and Anderson localization, Phys. Rev. Lett. 49 (1982), no. 8, 509–512. MR 669169, DOI 10.1103/PhysRevLett.49.509
- 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
- S. Jitomirskaya and W. Liu, Arithmetic spectral transitions for the Maryland model, Preprint, 2014.
- B. A. Khoruzhenko and L. A. Pastur, The localization of surface states: an exactly solvable model, Physics Reports, 288(1):109–126, 1997.
- M. Reed and B. Simon, Methods of Modern Mathematical Physics: Vol.: 1.: Functional Analysis, Academic press, 1972.
- 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
Additional Information
- Wencai Liu
- Affiliation: School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China
- Address at time of publication: Department of Mathematics, University of California, Irvine, California 92697-3875
- MR Author ID: 1030969
- ORCID: setImmediate$0.31799537312117976$2
- Email: liuwencai1226@gmail.com
- Received by editor(s): November 18, 2015
- Received by editor(s) in revised form: January 11, 2016
- Published electronically: August 5, 2016
- Communicated by: Michael Hitrik
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 5035-5047
- MSC (2010): Primary 11F72, 37A30, 47A10
- DOI: https://doi.org/10.1090/proc/13093
- MathSciNet review: 3556250