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1 References 1. Anderson, P.W.: Absence of diffusion in certain random lattices. Phys. Rev. 109, (1958) 2. Abrahams, E. (ed.): 50 Years of Anderson Localization. World Scientific, Singapore (2010); reprinted in Int. J. Mod. Phys. B 24(12 13) (2010) 3. Abrahams, E., Anderson, P.W., Licciardello, D.C., Ramakrishnan, T.C.: Scaling theory of localization: absence of quantum diffusion in two dimensions. Phys. Rev. Lett. 42, (1979) 4. Abu-Chacra, R., Anderson, P.W., Thouless, D.J.: A self consistent theory of localization. J. Phys. C 6, (1973) 5. Abu-Chacra, R., Anderson, P.W., Thouless, D.J.: Self consistent theory of localization. II. Localization near the band edges. J. Phys. C 7, (1974) 6. Aizenman, M.: Localization at weak disorder: some elementary bounds. Rev. Math. Phys. 06(special issue), (1994) 7. Aizenman, M., Molchanov, S.: Localization at large disorder and at extreme energies: an elementary derivation. Commun. Math. Phys. 157, (1993) 8. Aizenman, M., Warzel, S.: The canopy graph and level statistics for random operators on trees. Math. Phys. Anal. Geom. 9(4), (2007) 9. Aizenman, M., Warzel, S.: Localization bounds for multiparticle systems. Commun. Math. Phys. 290, (2009) 10. Aizenman, M., Warzel, S.: Complete dynamical localization in disordered quantum multiparticle systems. In: XVIth International Congress on Mathematical Physics, Prague, pp World Scientific (2010) 11. Aizenman, M., Warzel, S.: Extended states in a Lifshits tail regime for random Schrödinger operators on trees. Phys. Rev. Lett. 106, (2011) 12. Aizenman, M., Warzel, S.: Resonant delocalization for random Schrödinger operators on tree graphs. J. Eur. Math. Soc. (2011, to appear). Preprint, arxiv:math-ph/1104: Aizenman, M., Schenker, J.H., Friedrich, R.M., Hundertmark, D.: Finite-volume fractionalmoment criteria for Anderson localization. Commun. Math. Phys. 224, (2001) 14. Aizenman, M., Elgart, A., Naboko, S., Schenker, J.H., Stoltz, G.: Moment analysis for localization in random Schrödinger operators. Invent. Math. 163, (2006) 15. Aizenman, M., Sims, R., Warzel, S.: Stability of the absolutely continuous spectrum of random Schrödinger operators on tree graphs. Probab. Theory Relat. Fields 136, (2006) 16. Aizenman, M., Sims, R., Warzel, S.: Absolutely continuous spectra of quantum tree graphs with weak disorder. Commun. Math. Phys. 264, (2006) V. Chulaevsky and Y. Suhov, Multi-scale Analysis for Random Quantum Systems with Interaction, Progress in Mathematical Physics 65, DOI / , Springer Science+Business Media New York
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4 232 References 69. Chulaevsky, V., Suhov, Y.: Anderson localisation for an interacting two-particle quantum system on Z (2007). arxiv:math-ph/ Chulaevsky, V., Suhov, Y.: Eigenfunctions in a two-particle Anderson tight binding model. Commun. Math. Phys. 289, (2009) 71. Chulaevsky, V., Suhov, Y.: Multi-particle Anderson localisation: induction on the number of particles. Math. Phys. Anal. Geom. 12, (2009) 72. Chulaevsky, V., Boutet de Monvel, A., Suhov, Y.: Dynamical localization for a multiparticle model with an alloy-type external random potential. Nonlinearity 24(5), (2011) 73. Chulaevsky, V., Boutet de Monvel, A., Suhov, Y.: Multi-particle dynamical localization in a Euclidean space with a Gaussian random potential (in preparation) 74. Combes, J.-M., Thomas, L.: Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators. Commun. Math. Phys. 34, (1973) 75. Combes, J.-M., Hislop, P.D., Klopp, F.: An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators. Duke Math. J. 140(3), (2007) 76. Combes, J.-M., Germinet, F., Hislop, P.: Conductivity and the current current correlation measure. J. Phys. A 43, (2010) 77. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators. Springer, Berlin (1987) 78. Damanik, D., Stollmann, P.: Multi-scale analysis implies strong dynamical localization. Geom. Funct. Anal. 11(1), (2001) 79. Del Rio, R., Jitomirskaya, L., Last, Y., Simon, B.: Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization. J. Anal. Math. 69, (1996) 80. Dinaburg, E.I., Sinai, Y.G.: On the spectrum of a one-dimensional Schrödinger operator with a quasi-periodic potential. Funct. Anal. Appl. 9, 8 21 (1975) 81. Disertori, M., Kirsch, W., Klein, A., Klopp, F., Rivasseau, V: Random Schrödinger Operators. Panoramas et Synthèses, vol. 25. Société Mathématique de France, Paris (2008) 82. Ekanga, T.: On two-particle Anderson localization at low energies. C. R. Acad. Sci. Paris I 349(3 4), (2011) 83. Ekanga, T.: Anderson localization in the multi-particle tight-binding model at low energies or with weak interaction (2012). Preprint, arxiv:math-ph/ Elgart, A., Tautenhahn, M., Veselić, I.: Anderson localization for a class of models with a sign-indefinite single-site potential via fractional moment method. Ann. Henri Poincaré 12(8), (2010) 85. Eliasson, L.H.: Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146, (1992) 86. Eliasson, L.H.: Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum. Acta Math. 179, (1997) 87. Eliasson, L.H.: On the discrete one-dimensional quasi-periodic Schrödinger equation and other smooth quasi-periodic skew products. In: Hamiltonian Systems with Three or More Degrees of Freedom (S Agaró, 1995). NATO Advanced Science Institute Series, vol. 533, pp Kluwer, Dordrecht (1999) 88. Enss, V.: Asymptotic completeness for quantum-mechanical potential scattering. Short-range potentials. Commun. Math. Phys. 61, (1978) 89. Exner, P., Helm, M., Stollmann, P.: Localization on a quantum graph with a random potential on edges. Rev. Math. Phys. 19, (2007) 90. Exner, P., Keating, J.P., Kuchment, P., Sunada, T., Teplyaev, A. (eds.): Analysis on Graphs and Its Applications. Proceedings of Symposia in Pure Mathematics, vol. 77. American Mathematical Society, Providence (2008) 91. Figotin, A., Pastur, L.: An exactly solvable model of a multidimensional incommensurate structure. Commun. Math. Phys. 95, (1984) 92. Fischer, W., Leschke, H., Müller, P.: Spectral localization by Gaussian random potentials in multi-dimensional continuous space. J. Stat. Phys. 101(5/6), (2000)
5 References Fishman, S., Grempel, D., Prange, R.: Localization in a d-dimensional incommensurate structure. Phys. Rev. B 194, (1984) 94. Fleishman, L., Anderson, P.W.: Interactions and the Anderson transition. Phys. Rev. B 21, (1980) 95. Fröhlich, J., Spencer, T.: Absence of diffusion in the Anderson tight-binding model for large disorder or low energy. Commun. Math. Phys. 88, (1983) 96. Fröhlich, J., Martinelli, F., Scoppola, E., Spencer, T.: Constructive proof of localization in the Anderson tight-binding model. Commun. Math. Phys. 101, (1985) 97. Fröhlich, J., Spencer, T., Wittwer, P.: Localization for a class of one-dimensional quasiperiodic Schrödinger operators. Commun. Math. Phys. 132, 5 25 (1990) 98. Germinet, F., De Bièvre, S.: Dynamical localization for discrete and continuous random Schrödinger operators. Commun. Math. Phys. 194, (1998) 99. Germinet, F., Klein, A.: Bootstrap multi-scale analysis and localization in random media. Commun. Math. Phys. 222, (2001) 100. Germinet, F., Klein, A.: A comprehensive proof of localization for continuous Anderson models with singular random potentials. J. Eur. Math. Soc. (2011, to appear). arxiv:math-ph/ Goldsheid, I.Y., Molchanov, S.A.: On Mott s problem. Sov. Math. Dokl. 17, (1976) 102. Goldsheid, I.Y., Molchanov, S.A., Pastur, L.A.: A pure point spectrum of the one-dimensional Schrödinger operator. Funct. Anal. Appl. 11, 1 10 (1977) 103. Gordon, A.Y.: On the point spectrum of the one-dimensional Schrödinger operator. (Russian) Uspekhi Matem. Nauk 31, (1976) 104. Gordon, A.Y., Jitomirskaya, S., Last, Y., Simon, B.: Dualityand singular continuous spectrum in the almost Mathieu equation. Acta Math. 178, (1997) 105. Gornyi, I.V., Mirlin, A.D., Polyakov, D.G.: Interacting electrons in disordered wires: Anderson localization and low-temperature transport. Phys. Rev. Lett. 95, (2005) 106. Graf, G.M., Vaghi, A.: A remark on the estimate of a determinant by Minami. Lett. Math. Phys. 79, (2007) 107. Grempel, D., Fishman, S., Prange, R.: Localization in an incommensurate potential: an exactly solvable model. Phys. Rev. Lett. 49, 833 (1982) 108. Harper, P.G.: Single band motion of conducting electrons in a uniform magnetic field. Proc. Phys. Soc. Lond. A 68, (1955) 109. Hofstadter, D.R.: Energy levels and wavefunctions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, (1976) 110. Hundertmark, D.: A short introduction to Anderson localization. In: Mörters, P., et al. (ed.) Analysis and Stochastics of Growth Processes an Interface Models. Oxford University Press (2008) Jitomirskaya, S.Y.: Metal-insulator transition for the almost Mathieu operator. Ann. Math. 150, (1999) 112. Kato, T.: Perturbation Theory for Linear Operators. Springer, New York (1976) 113. Kirsch, W.: An invitation to random Schrödinger operators. With an appendix by F. Klopp, in Ref. [81], pp (2008) 114. Kirsch, W.: A Wegner estimate for multi-particle random Hamiltonians. Zh. Mat. Fiz. Anal. Geom. 4, (2008) 115. Kirsch, W., Stollmann, P., Stolz, G.: Anderson localization for random Schrödinger operators with long range interactions. Commun. Math. Phys. 195, (1998) 116. Klein, A.: Absolutely continuous spectrum in the Anderson model on the Bethe lattice. Math. Res. Lett. 1, (1994) 117. Klein, A.: Extended states in the Anderson model on the Bethe lattice. Adv. Math. 133(1), (1998) 118. Klein, A.: Multiscale analysis and localization of random operators. In: Ref. [81], pp Klein, A., Molchanov, S.: Simplicity of eigenvalues in the Anderson model. J. Stat. Phys. 122, (2006)
6 234 References 120. Klopp, F., Zenk, H.: The integrated density of states for an interacting multiparticle homogeneous model and applications to the Anderson model. Adv. Math. Phys. 2009, 1 15 (2009). Art. ID Kohn, W.: Theory of the insulating state. Phys. Rev. 133, A171 A181 (1964) 122. Kravchenko, S.V., Sarachik, M.P.: A metal insulator transition in 2D: established facts and open questions. Preprint, arxiv:math-ph/ ; also, In: Abrahams, E. (ed.) 50 Years of Anderson Localization, p World Scientific, Singapore (2010); reprinted in Int. J. Mod. Phys. B 24, (2010) 123. Kunz, H., Souillard, B.: Sur le spectre des opérateurs aux différences finies aléatoires. Commun. Math. Phys. 78, (1980) 124. Kunz, H., Souillard, B.: The localization transition on the Bethe lattice. J. Phys. Lett. 44, (1983) 125. Lagendiik, A., van Tiggelen, B., Wiersma, D.S.: Fifty years of Anderson localization. Phys. Today 62, (2009) 126. Lifshitz, I.M.: Structure of the energy spectrum of the impurity bands in disordered solids. Sov. Phys. JETP 17, (1963) 127. Lifshitz, I.M.: The energy spectrum of disordered systems. Adv. Phys. 13, (1964) 128. Lifshitz, I.M., Gredescul, S.A., Pastur, L.A.: Introduction to the Theory of Disordered Systems. Wiley, New York (1988) 129. Martinelli, F.: A quantum particle in a hierarchical potential: a first step towards the analysis of complex quantum systems. In: Phénomènes critiques, systèmes aléatoires, théories de jauge, Les Houches, 1984, pp North-Holland, Amsterdam (1986) 130. Martinelli, F., Holden, H.: On absence of diffusion near the bottom of the spectrum for a random Schrödinger operator on L 2.R d /. Commun. Math. Phys. 93, (1984) 131. Martinelli, F., Scoppola, E.: Absence of absolutely continuous spectrum in the Anderson model for large disorder or low energy. In: Infinite-Dimensional Analysis and Stochastic Processes, Bielefeld, Research Notes in Mathematics, vol. 124, pp Pitman, Boston (1983) 132. Martinelli, F., Scoppola, E.: Remark on the absence of absolutely continuous spectrum for d-dimensional Schrödinger oerators with random potential for large disorder or low energy. Commun. Math. Phys. 97, (1985) 133. Minami, N.: Local fluctuation of the spectrum of a multidimensional Anderson tight-binding model. Commun. Math. Phys. 177, (1996) 134. Molchanov, S.A.: Structure of eigenfunctions of one-dimensional unordered structures. (Russian) Math. USSR Izv. 42, (1978) 135. Molchanov, S.A.: The local structure of the spectrum of the one-dimensional Schrödinger operator. Commun. Math. Phys. 78, (1981) 136. Moser, J., Pöschel, J.: An extension of a result by Dinaburg and Sinai on quasi-periodic potentials. Comment. Math. Helv. 59, (1984) 137. Mott, N.F.: Metal insulator transition. Rev. Mod. Phys. 40, (1968) 138. Mott, N.F., Twose, W.D.: The theory of impurity conditions. Adv. Phys. 10, (1961) 139. Nakano, F.: The repulsion between localization centers in the Anderson model. Commun. Math. Phys. 123(4), (2006) 140. Nakano, F.: Distribution of localization centers in some discrete random systems. Rev. Math. Phys. 19, (2007) 141. Novikov, S.P.: Periodic problem for the Korteveg de Vries equation. Funct. Anal. Appl. 8, (1974) 142. Pankrashkin, K.: Quasiperiodic surface Maryland models on quantum graphs. J. Phys. A 42, (2009) 143. Pastur, L., Figotin, A.: An exactly solvable model of a multidimensional incommensurate structure. Commun. Math. Phys. 95, (1984) 144. Pastur, L., Figotin, A.: Spectra of Random and Almost-Periodic Operators. Springer, Berlin (1992)
7 References Puig, J.: Cantor spectrum for the almost Mathieu operator. Commun. Math. Phys. 244, (2004) 146. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. 1. Academic, New York (1980) 147. Ruelle, D.: A remark on bound states in potential scattering theory. Nouvo Cimento 61A, (1969) 148. Sabri, M.: Anderson localization for a multi-particle quantum graph. Rev. Math. Phys. (2012, to appear). Preprint, arxiv:math-ph/ Shepelyansky, D.L.: Coherent propagation of two interacting particles in a random potential. Phys. Rev. Lett. 73, (1994) 150. Shnol, I.: On the behaviour of the Schrödinger equation. (Russian) Mat. Sb. 42, (1957) 151. Simon, B.: Schrödinger semigroups. Bull. Am. Math. Soc. 7, (1983) 152. Simon, B.: Almost periodic Schrödinger operators. IV: The Maryland model. Ann. Phys. 159, (1985) 153. Simon, B., Wolff, T.: Singular continuous spectrum under rank-one perturbations and localization for random Hamiltonians. Commun. Pure Appl. Math. 39, (1986) 154. Sinai, Y.G.: Anderson localization for one-dimensional difference Schrödinger operator with quasi-periodic potential. J. Stat. Phys. 46, (1987) 155. Spencer, T.: The Schrödinger equation with a random potential. A mathematical review. In: Critical Phenomena, Random Systems, Gauge Theories. Proc. Summer Sch. Theor. Phys. Sess., vol. 43, pp Les Houches, France 1984, Pt. 2 (1986) 156. Spencer, T.: Localization for random and quasi-periodic potentials. J. Stat. Phys. 51, (1988) 157. Stollmann, P.: Wegner estimates and localization for continuum Anderson models with some singular distributions. Arch. Math. 75, (2000) 158. Stollmann, P.: Caught by Disorder. Birkhäuser, Boston (2001) 159. Suhov, Y., Kelbert, M.: Probability and Statistics by Example. Markov Chains: A Primer in Random Processes and Their Applications, vol. 2. Cambridge University Press, Cambridge (2007) 160. von Dreifus, H.: On effect of randomness in ferromagneic models and Schrödinger operators. PhD dissertation, New York University, New York (1987) 161. von Dreifus, H., Klein, A.: A new proof of localization in the Anderson tight-binding model. Commun. Math. Phys. 124, (1989) 162. von Dreifus, H., Klein, A.: Localization for random Schrödinger operators with correlated potentials. Commun. Math. Phys. 140, (1991) 163. Wegner, F.: Bounds on the density of states of disordered systems. Z. Phys. B44, 9 15 (1981)
8 Index A Assumption A, 28 Assumption B, 28 Assumption C, 169 Assumption U, 143 C Carmona s argument, 155 center of localization, 123 Combes Thomas estimate, 62 combined multi-particle induction, 171 completely non-resonant (E-CNR) multiparticle cube, 177 completely non-resonant (E-CNR) singleparticle cube, 48 conditional Wegner bound, 156 D density of states, DoS, 50 diagonally monotone function, 45 distant n-particle cubes; 11nL-distant n-particle cubes, 198 double singularity for multi-particle systems (DS.N; n; m; p k ;L k ), 146 double singularity for multi-particle systems, modified.f DS:N;n;m;p0 ;L 0 /, 196 double singularity.ds:i;m;p k ;L k / for single-particle systems, 111 dynamical localization, 15, 18, 20, 41, 121, 148 E eigenfunction (EF) correlator, 41 eigenvalue concentration (EVC) bounds, 43 external random potential, 28 F finite range of interaction, 174 fixed-energy analysis, 33, 34 fixed-energy analysis, multi-particle, 176 Fractional moment method, FMM, 18 fully interactive (FI) cube, 174 G geometric resolvent equation, 55, 56 geometric resolvent inequality (GRI), 55 geometric resolvent inequality (GRI) for eigenfunctions, 57 I independent identically distributed (IID) potential, 27 integrated density of states (IDoS), 214, interaction, 139, 143, 174 J jump Markov process, 153 L lattice Schrödinger operator (LSO), 7 localization center, 123 M mass, 34 Minami bound, 52 Molchanov s path integral formula, 152 MPMSA induction: the inductive step, fixed energy, 185 V. Chulaevsky and Y. Suhov, Multi-scale Analysis for Random Quantum Systems with Interaction, Progress in Mathematical Physics 65, DOI / , Springer Science+Business Media New York
9 238 Index MPMSA induction: the inductive step, variable energy, 197 MPMSA induction: the initial step, fixed energy, 177 MPMSA induction: the initial step, variable energy, 196 multi-particle MSA (MPMSA), 137 Multi-scale analysis (MSA), 8 N non-partially singular cube,.m;i;k/-nonpartially singular (.m;i;k/-nps), 108 non-resonant (E-NR) multi-particle cube, 177 non-resonant (E-NR) single-particle cube, 48 non-singular (.E; m/-ns) multi-particle cube, 143 non-singular (.E; m/-ns) single-particle cube, 33 O one-volume Stollmann bound for multi-particle systems, 162 P pair of distant FI cubes, 182 pair of FI cubes, 175, 197 pair of multi-particle cubes (mixed), 197 pair of PI cubes, 197 partially interactive (PI) cube, 174 partially resonant (E-PR) multi-particle cube, 177 partially resonant (E-PR) single-particle cube, 48 partially singular cube,.m;i;k/-partially singular (.m;i;k/-ps), 108 Q quantum graphs, 5, 19, 226 R radial descent bounds, 102 resonant (E-R) single-particle cube, 48 resonant (E-R) multi-particle cube, 177 S separability of cubes, 144 singular (.E; m/-s) multi-particle cube, 143 singular (.E; m/-s) single-particle cube, 33 singularity for multi-particle systems (SS.N; E; m; p k ;L k ), 145 spectral localization, 18 Stollmann s bound, 47 Stollmann s lemma, 45 strong dynamical localization, 19 subharmonic (.l; q/-subharmonic), 75 T two-volume Stollmann bound for multi-particle systems, 163 V variable-energy analysis, 37 variable-energy analysis, multi-particle, 195 W weak separability of cubes, 167 Wegner s bound, 43, 156, 164
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