Book Review
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MathSciNet review:
3686330
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Book Information:
Authors:
Michael Aizenman and
Simone Warzel
Title:
Random operators: disorder effects on quantum spectra and dynamics
Additional book information:
Graduate Studies in Mathematics, Vol. 168,
American Mathematical Society,
2015,
xiv+326 pp.,
ISBN 978-1-4704-1913-4,
US $79.00. Individual member price $63.20.
H. Abdul-Rahman, B. Nachtergaele, R. Sims and G. Stolz, Mathematical results on many-body localization in the disordered XY spin chain, Preprint 2016, arXiv:1610.01939
Michael Aizenman, Localization at weak disorder: some elementary bounds, Rev. Math. Phys. 6 (1994), no. 5A, 1163–1182. Special issue dedicated to Elliott H. Lieb. MR 1301371, DOI 10.1142/S0129055X94000419
M. Aizenman and G. M. Graf, Localization bounds for an electron gas, J. Phys. A 31 (1998), no. 32, 6783–6806. MR 1715186, DOI 10.1088/0305-4470/31/32/004
Michael Aizenman and Stanislav Molchanov, Localization at large disorder and at extreme energies: an elementary derivation, Comm. Math. Phys. 157 (1993), no. 2, 245–278. MR 1244867
P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109 (1958), 1492–1505.
D. M. Basko, I. L. Aleiner and B. L. Altshuler, Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states, Annals of Phyiscs 321 (2006), 1126–1205.
Jean Bourgain and Carlos E. Kenig, On localization in the continuous Anderson-Bernoulli model in higher dimension, Invent. Math. 161 (2005), no. 2, 389–426. MR 2180453, DOI 10.1007/s00222-004-0435-7
René Carmona and Jean Lacroix, Spectral theory of random Schrödinger operators, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1102675, DOI 10.1007/978-1-4612-4488-2
Michael Demuth and M. Krishna, Determining spectra in quantum theory, Progress in Mathematical Physics, vol. 44, Birkhäuser Boston, Inc., Boston, MA, 2005. MR 2159561
Eugene Feenberg, A note on perturbation theory, Phys. Rev. (2) 74 (1948), 206–208. MR 26432
Jürg Fröhlich and Thomas Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Comm. Math. Phys. 88 (1983), no. 2, 151–184. MR 696803
François Germinet and Abel Klein, A comprehensive proof of localization for continuous Anderson models with singular random potentials, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 1, 53–143. MR 2998830, DOI 10.4171/JEMS/356
Dirk Hundertmark, A short introduction to Anderson localization, Analysis and stochastics of growth processes and interface models, Oxford Univ. Press, Oxford, 2008, pp. 194–218. MR 2603225, DOI 10.1093/acprof:oso/9780199239252.003.0009
John Z. Imbrie, On many-body localization for quantum spin chains, J. Stat. Phys. 163 (2016), no. 5, 998–1048. MR 3493184, DOI 10.1007/s10955-016-1508-x
John Z. Imbrie, Multi-scale Jacobi method for Anderson localization, Comm. Math. Phys. 341 (2016), no. 2, 491–521. MR 3440194, DOI 10.1007/s00220-015-2522-6
Werner Kirsch, An invitation to random Schrödinger operators, Random Schrödinger operators, Panor. Synthèses, vol. 25, Soc. Math. France, Paris, 2008, pp. 1–119 (English, with English and French summaries). With an appendix by Frédéric Klopp. MR 2509110
Hervé Kunz and Bernard Souillard, Sur le spectre des opérateurs aux différences finies aléatoires, Comm. Math. Phys. 78 (1980/81), no. 2, 201–246 (French, with English summary). MR 597748
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
Barry Simon, Spectral analysis of rank one perturbations and applications, Mathematical quantum theory. II. Schrödinger operators (Vancouver, BC, 1993) CRM Proc. Lecture Notes, vol. 8, Amer. Math. Soc., Providence, RI, 1995, pp. 109–149. MR 1332038, DOI 10.1090/crmp/008/04
Barry Simon and Tom Wolff, Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Comm. Pure Appl. Math. 39 (1986), no. 1, 75–90. MR 820340, DOI 10.1002/cpa.3160390105
Peter Stollmann, Caught by disorder, Progress in Mathematical Physics, vol. 20, Birkhäuser Boston, Inc., Boston, MA, 2001. Bound states in random media. MR 1935594, DOI 10.1007/978-1-4612-0169-4
Günter Stolz, An introduction to the mathematics of Anderson localization, Entropy and the quantum II, Contemp. Math., vol. 552, Amer. Math. Soc., Providence, RI, 2011, pp. 71–108. MR 2868042, DOI 10.1090/conm/552/10911
References
- H. Abdul-Rahman, B. Nachtergaele, R. Sims and G. Stolz, Mathematical results on many-body localization in the disordered XY spin chain, Preprint 2016, arXiv:1610.01939
- Michael Aizenman, Localization at weak disorder: some elementary bounds, Rev. Math. Phys. 6 (1994), no. 5A, 1163–1182. MR 1301371, DOI 10.1142/S0129055X94000419
- M. Aizenman and G. M. Graf, Localization bounds for an electron gas, J. Phys. A 31 (1998), no. 32, 6783–6806. MR 1715186, DOI 10.1088/0305-4470/31/32/004
- Michael Aizenman and Stanislav Molchanov, Localization at large disorder and at extreme energies: an elementary derivation, Comm. Math. Phys. 157 (1993), no. 2, 245–278. MR 1244867
- P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109 (1958), 1492–1505.
- D. M. Basko, I. L. Aleiner and B. L. Altshuler, Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states, Annals of Phyiscs 321 (2006), 1126–1205.
- Jean Bourgain and Carlos E. Kenig, On localization in the continuous Anderson-Bernoulli model in higher dimension, Invent. Math. 161 (2005), no. 2, 389–426. MR 2180453, DOI 10.1007/s00222-004-0435-7
- René Carmona and Jean Lacroix, Spectral theory of random Schrödinger operators, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1102675, DOI 10.1007/978-1-4612-4488-2
- Michael Demuth and M. Krishna, Determining spectra in quantum theory, Progress in Mathematical Physics, vol. 44, Birkhäuser Boston, Inc., Boston, MA, 2005. MR 2159561
- Eugene Feenberg, A note on perturbation theory, Physical Rev. (2) 74 (1948), 206–208. MR 0026432
- Jürg Fröhlich and Thomas Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Comm. Math. Phys. 88 (1983), no. 2, 151–184. MR 696803
- François Germinet and Abel Klein, A comprehensive proof of localization for continuous Anderson models with singular random potentials, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 1, 53–143. MR 2998830, DOI 10.4171/JEMS/356
- Dirk Hundertmark, A short introduction to Anderson localization, Analysis and stochastics of growth processes and interface models, Oxford Univ. Press, Oxford, 2008, pp. 194–218. MR 2603225, DOI 10.1093/acprof:oso/9780199239252.003.0009
- John Z. Imbrie, On many-body localization for quantum spin chains, J. Stat. Phys. 163 (2016), no. 5, 998–1048. MR 3493184, DOI 10.1007/s10955-016-1508-x
- John Z. Imbrie, Multi-scale Jacobi method for Anderson localization, Comm. Math. Phys. 341 (2016), no. 2, 491–521. MR 3440194, DOI 10.1007/s00220-015-2522-6
- Werner Kirsch, An invitation to random Schrödinger operators, Random Schrödinger operators, Panor. Synthèses, vol. 25, Soc. Math. France, Paris, 2008, pp. 1–119 (English, with English and French summaries). With an appendix by Frédéric Klopp. MR 2509110
- Hervé Kunz and Bernard Souillard, Sur le spectre des opérateurs aux différences finies aléatoires, Comm. Math. Phys. 78 (1980/81), no. 2, 201–246 (French, with English summary). MR 597748
- 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
- Barry Simon, Spectral analysis of rank one perturbations and applications, Mathematical quantum theory. II. Schrödinger operators (Vancouver, BC, 1993) CRM Proc. Lecture Notes, vol. 8, Amer. Math. Soc., Providence, RI, 1995, pp. 109–149. MR 1332038
- Barry Simon and Tom Wolff, Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Comm. Pure Appl. Math. 39 (1986), no. 1, 75–90. MR 820340, DOI 10.1002/cpa.3160390105
- Peter Stollmann, Caught by disorder : Bound states in random media, Progress in Mathematical Physics, vol. 20, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1935594, DOI 10.1007/978-1-4612-0169-4
- Günter Stolz, An introduction to the mathematics of Anderson localization, Entropy and the quantum II, Contemp. Math., vol. 552, Amer. Math. Soc., Providence, RI, 2011, pp. 71–108. MR 2868042, DOI 10.1090/conm/552/10911
Review Information:
Reviewer:
Günter Stolz
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294
Email:
stolz@uab.edu
Journal:
Bull. Amer. Math. Soc.
54 (2017), 347-353
DOI:
https://doi.org/10.1090/bull/1565
Published electronically:
December 20, 2016
Review copyright:
© Copyright 2016
American Mathematical Society
