Abstract
The present note reviews recent results on resonances for one-dimensional quantum ergodic systems constrained to a large box (see [14]). We restrict ourselves to one-dimensional models in the discrete case. We consider two type of ergodic potentials on the half-axis, periodic potentials and random potentials. For both models, we describe the behavior of the resonances near the real axis for a large typical sample of the potential. In both cases, the linear density of their real parts is given by the density of states of the full ergodic system. While in the periodic case, the resonances distribute on a nice analytic curve (once their imaginary parts are suitably renormalized), in the random case, the resonances (again after suitable renormalization of both the real and imaginary parts) form a two-dimensional Poisson cloud.
Mathematics Subject Classification (2010). 47B80, 47H40, 60H25, 82B44, 35B34.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
F. Barra and P. Gaspard. Scattering in periodic systems: from resonances to band structure. J. Phys. A, 32(18):3357-3375, 1999.
R. Carmona and J. Lacroix. Spectral theory of random Schrödinger operators. Probability and its Applications. Birkhäuser Boston Inc., Boston, MA, 1990.
A. Comtet and C. Texier. On the distribution of the Wigner time delay in one- dimensional disordered systems. J. Phys. A, 30(23):8017-8025, 1997.
H.L. Cycon, R.G. Froese, W. Kirsch, and B. Simon. Schrödinger operators with application to quantum mechanics and global geometry. Texts and Monographs in Physics. Springer-Verlag, Berlin, study edition, 1987.
W.G. Faris and W.J. Tsay. Scattering of a wave packet by an interval of random medium. J. Math. Phys., 30(12):2900-2903, 1989.
W.G. Faris and W.J. Tsay. Time delay in random scattering. SIAM J. Appl. Math., 54(2):443-455, 1994.
F. Germinet and F. Klopp. Spectral statistics for random Schrödinger operators in the localized regime. Ar**v http://arxiv.org/abs/1011.1832, 2010.
F. Germinet and F. Klopp. Enhanced Wegner and Minami estimates and eigenvalue statistics of random Anderson models at spectral edges. Ar**v http://arxiv.org/abs/1111.1505 , 2011.
F. Germinet and F. Klopp. Spectral statistics for the discrete Anderson model in the localized regime. In N. Minami, editor, Spectra of random operators and related topics, 2011. To appear. Ar**v http://arxiv.org/abs/1004.1261.
F.W. King. Hilbert transforms. Vol. 1, volume 124 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2009.
W. Kirsch. An invitation to random Schrödinger operators. In Random Schrödinger operators, volume 25 of Panor. Synthèses, pages 1-119. Soc. Math. France, Paris, 2008. With an appendix by Frédéric Klopp.
F. Klopp. Decorrelation estimates for the discrete Anderson model. Comm. Math. Phys., 303(1):233-260, 2011.
F. Klopp. Asymptotic ergodicity of the eigenvalues of random operators in the localized phase. To appear in PTRF, Ar**v: http://fr.arxiv.org/abs/1012.0831, 2010.
F. Klopp. Resonances for large one-dimensional “ergodic” systems. In preparation, 2011.
T. Kottos. Statistics of resonances and delay times in random media: beyond random matrix theory. J. Phys. A, 38(49):10761-10786, 2005.
H. Kunz and B. Shapiro. Resonances in a one-dimensional disordered chain. J. Phys. A, 39(32):10155-10160, 2006.
H. Kunz and B. Shapiro. Statistics of resonances in a semi-infinite disordered chain. Phys. Rev. B, 77(5):054203, Feb. 2008.
I.M. Lifshits, S.A. Gredeskul, and L.A. Pastur. Introduction to the theory of disordered systems. A Wiley-Interscience Publication. John Wiley & Sons Inc., New York, 1988. Translated from the Russian by Eugene Yankovsky [E.M. Yankovskii].
M. Merkli and I.M. Sigal. A time-dependent theory of quantum resonances. Comm. Math. Phys., 201(3):549-576, 1999.
N. Minami. Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Comm.. Math. Phys., 177(3):709-725, 1996.
S.A. Molchanov. The local structure of the spectrum of a random one-dimensional Schrödinger operator. Trudy Sem. Petrovsk., (8):195-210, 1982.
L.A. Pastur and A. Figotin. Spectra of random and almost-periodic operators, volume 297 of Grund,lehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1992.
B. Pavlov. Nonphysical sheet for perturbed Jacobian matrices. Algebra i Analiz, 6(3):185-199, 1994.
G. Teschl. Jacobi operators and completely integrable nonlinear lattices, volume 72 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2000.
C. Texier and A. Comtet. Universality of the Wigner Time Delay Distribution for One-Dimensional Random Potentials. Physical Review Letters, 82:4220-4223, May 1999.
M. Titov and Y.V. Fyodorov. Time-delay correlations and resonances in one- dimensional disordered systems. Phys. Rev. B, 61(4):R2444-R2447, Jan 2000.
P. van Moerbeke. The spectrum of Jacobi matrices. Invent. Math., 37(1):45-81, 1976.
M. Zworski. Quantum resonances and partial differential equations. In Proceedings of the International Congress of Mathematicians, Vol. III (Bei**g, 2002), pages 243-252, Bei**g, 2002. Higher Ed. Press.
M. Zworski. Resonances in physics and geometry. Notices Amer. Math. Soc., 46(3):319-328, 1999.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Basel
About this paper
Cite this paper
Klopp, F. (2012). Resonances for “Large” Ergodic Systems in One Dimension: A Review. In: Benguria, R., Friedman, E., Mantoiu, M. (eds) Spectral Analysis of Quantum Hamiltonians. Operator Theory: Advances and Applications, vol 224. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0414-1_9
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0414-1_9
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0413-4
Online ISBN: 978-3-0348-0414-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)