Single-Particle MSA Techniques

Chapter 2 prepares the ground for the analysis of interacting particle systems carried out in Part II. The authors introduce here the principal technical tools of the single-particle multi-scale analysis (MSA), developed over the last thirty years by the mathematical community. The analytical tools of the so-called variable-energy MSA, developed in late 1980s, are streamlined and complemented by a simpler, fixed-energy approach. A simple and comprehensive derivation of the spectral and strong dynamical localization from the fixed-energy MSA, suitable for adaptations to interacting systems, is presented for the first time in mathematical literature.

[1]  M. Sarachik,et al.  a Metal-Insulator Transition in 2D:. Established Facts and Open Questions , 2010, 1003.2968.

[2]  M. Aizenman,et al.  Localization Bounds for Multiparticle Systems , 2008, 0809.3436.

[3]  F. Wegner.  Bounds on the density of states in disordered systems , 1981 .

[4]  A. Avila,et al.  Almost localization and almost reducibility , 2008, 0805.1761.

[5]  M. Aizenman,et al.  The Canopy Graph and Level Statistics for Random Operators on Trees , 2006, math-ph/0607021.

[6]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[7]  A. Klein,et al.  A new proof of localization in the Anderson tight binding model , 1989 .

[8]  J. Fröhlich,et al.  Absence of diffusion in the Anderson tight binding model for large disorder or low energy , 1983 .

[9]  Kristian Bjerklöv.  Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrödinger equations , 2005, Ergodic Theory and Dynamical Systems.

[10]  P. Bougerol,et al.  Products of Random Matrices with Applications to Schrödinger Operators , 1985 .

[11]  D. Basko,et al.  Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states , 2005, cond-mat/0506617.

[12]  J. Combes,et al.  Conductivity and the current–current correlation measure , 2010, 1008.5344.

[13]  F. Martinelli,et al.  Remark on the absence of absolutely continuous spectrum ford-dimensional Schrödinger operators with random potential for large disorder or low energy , 1985 .

[14]  P. Anderson,et al.  Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions , 1979 .

[15]  P. Stollmann,et al.  Multi-scale analysis implies strong dynamical localization , 1999, math-ph/9912002.

[16]  Correlation Estimates in the Anderson Model , 2007, math-ph/0703058.

[17]  Kristian Bjerklöv.  Positive Lyapunov Exponent and Minimality for the Continuous 1-d Quasi-Periodic Schrödinger Equation with Two Basic Frequencies , 2007 .

[18]  P. G. Harper,et al.  Single Band Motion of Conduction Electrons in a Uniform Magnetic Field , 1955 .

[19]  AN OPTIMAL WEGNER ESTIMATE AND ITS APPLICATION TO THE GLOBAL CONTINUITY OF THE INTEGRATED DENSITY OF STATES FOR RANDOM SCHR , 2006, math-ph/0605029.

[20]  F. Delyon,et al.  Purely absolutely continuous spectrum for almost Mathieu operators , 1989 .

[21]  L. Pastur.  On the Schrödinger equation with a random potential , 1971 .

[22]  F. Germinet,et al.  Dynamical Localization for Discrete and Continuous Random Schrr Odinger Operators , 1997 .

[23]  J. Bourgain.  Anderson Localization for Quasi-Periodic Lattice Schrödinger Operators on $${\mathbb{Z}}^{d}$$ , d Arbitrary , 2007 .

[24]  J. Moser,et al.  An extension of a result by Dinaburg and Sinai on quasi-periodic potentials , 1984 .

[25]  B. Simon,et al.  Duality and singular continuous spectrum in the almost Mathieu equation , 1997 .

[26]  Nariyuki Minami,et al.  Local fluctuation of the spectrum of a multidimensional Anderson tight binding model , 1996 .

[27]  Jean Bourgain,et al.  Green's Function Estimates for Lattice Schrödinger Operators and Applications. , 2004 .

[28]  V. Chulaevsky.  A Wegner-type Estimate for Correlated Potentials , 2008 .

[29]  Hervé Kunz,et al.  Sur le spectre des opérateurs aux différences finies aléatoires , 1980 .

[30]  Michael Aizenman,et al.  On Bernoulli decompositions for random variables, concentration bounds, and spectral localization , 2007 .

[31]  I. Lifshitz,et al.  The energy spectrum of disordered systems , 1964 .

[32]  A. B. D. Monvel,et al.  Dynamical localization for a multi-particle model with an alloy-type external random potential , 2011 .

[33]  Stability of the Absolutely Continuous Spectrum of Random Schrödinger Operators on Tree Graphs , 2005, math-ph/0502006.

[34]  Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder , 2005, math-ph/0504039.

[35]  A. Klein,et al.  Bootstrap Multiscale Analysis and Localization¶in Random Media , 2001 .

[36]  G. Stolz,et al.  Anderson Localization for Random Schrödinger Operators with Long Range Interactions , 1998 .

[37]  T. Ekanga.  On two-particle Anderson localization at low energies , 2011, 1203.1207.

[38]  S. A. Molčanov,et al.  The local structure of the spectrum of the one-dimensional Schrödinger operator , 1981 .

[39]  P. Anderson.  Absence of Diffusion in Certain Random Lattices , 1958 .

[40]  Walter Kohn,et al.  THEORY OF THE INSULATING STATE , 1964 .

[41]  From fixed-energy MSA to dynamical localization: A continuing quest for elementary proofs , 2012, 1205.5763.

[42]  A. Elgart,et al.  Anderson Localization for a Class of Models with a Sign-Indefinite Single-Site Potential via Fractional Moment Method , 2010, 1011.5648.

[43]  V. Chulaevsky.  On the regularity of the conditional distribution of the sample mean , 2013, 1304.6913.

[44]  P. Anderson,et al.  Interactions and the Anderson transition , 1980 .

[45]  P. Stollmann.  Wegner estimates and localization for continuum Anderson models with some singular distributions , 2000 .

[46]  A. Klein.  Absolutely Continuous Spectrum in the Anderson Model on the Bethe Lattice , 1994 .

[47]  Nevill Mott,et al.  The theory of impurity conduction , 1961 .

[48]  B. Simon,et al.  Cantor spectrum for the almost Mathieu equation , 1982 .

[49]  A. Klein.  Extended States in the Anderson Model on the Bethe Lattice , 1998 .

[50]  Hartmut Schwetlick,et al.  Analysis and Stochastics of Growth Processes and Interface Models , 2008 .

[51]  LOCALIZATION ON A QUANTUM GRAPH WITH A RANDOM POTENTIAL ON THE EDGES , 2006, math-ph/0612087.

[52]  J. Bourgain.  An Approach to Wegner’s Estimate Using Subharmonicity , 2009 .

[53]  L. H. Eliasson,et al.  Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation , 1992 .

[54]  René Carmona,et al.  Anderson localization for Bernoulli and other singular potentials , 1987 .

[55]  The localization transition on the Bethe lattice , 1983 .

[56]  Y. Suhov,et al.  Multi-particle Anderson Localisation: Induction on the Number of Particles , 2008, 0811.2530.

[57]  Michael Aizenman,et al.  Moment analysis for localization in random Schrödinger operators , 2003, math-ph/0308023.

[58]  V. Enss.  Asymptotic completeness for quantum mechanical potential scattering , 1978 .

[59]  Y. Sinai,et al.  Anderson localization for the 1-D discrete Schrödinger operator with two-frequency potential , 1989 .

[60]  Mostafa Sabri.  Anderson Localization for a Multi-Particle Quantum Graph , 2012, 1201.6247.

[61]  T. Spencer.  Localization for random and quasiperiodic potentials , 1988 .

[62]  S. Aubry.  The New Concept of Transitions by Breaking of Analyticity in a Crystallographic Model , 1978 .

[63]  D. Hofstadter.  Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields , 1976 .

[64]  Jean Bourgain,et al.  Anderson localization for Schrödinger operators on Z2 with quasi-periodic potential , 2002 .

[65]  J. Fröhlich,et al.  Localization for a class of one dimensional quasi-periodic Schrödinger operators , 1990 .

[66]  On nonperturbative localization with quasi-periodic potential , 2000, math-ph/0011053.

[67]  O. Bratteli.  Operator Algebras And Quantum Statistical Mechanics , 1979 .

[68]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[69]  Spectral Localization by Gaussian Random Potentials in Multi-Dimensional Continuous Space , 1999, math-ph/9912025.

[70]  M. Aizenman,et al.  Resonant delocalization for random Schr\"odinger operators on tree graphs , 2011, 1104.0969.

[71]  Svetlana Ya. Jitomirskaya.  Metal-insulator transition for the almost Mathieu operator , 1999 .

[72]  D. Grempel,et al.  Localization in an Incommensurate Potential: An Exactly Solvable Model , 1982 .

[73]  B. Simon,et al.  Singular continuous spectrum under rank one perturbations and localization for random hamiltonians , 1986 .

[74]  D. Thouless,et al.  Self-consistent theory of localization. II. Localization near the band edges , 1974 .

[75]  B. Simon,et al.  Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization , 1996 .

[76]  L. Eliasson.  On The Discrete One-Dimensional Quasi-Periodic Schrödinger Equation and Other Smooth Quasi-Periodic Skew Products , 1999 .

[77]  M. Brin.  Topological transitivity of one class of dynamic systems and flows of frames on manifolds of negative curvature , 1975 .

[78]  V. Chulaevsky.  Direct Scaling Analysis of Localization in Single-Particle Quantum Systems on Graphs with Diagonal Disorder , 2012 .

[79]  M. Artin,et al.  Société Mathématique de France , 1994 .

[80]  V. Chulaevsky.  Anderson localization for generic deterministic operators , 2011, 1101.4892.

[81]  F. Martinelli,et al.  Constructive proof of localization in the Anderson tight binding model , 1985 .

[82]  J. Bourgain,et al.  Absolutely continuous spectrum for 1D quasiperiodic operators , 2002 .

[83]  V. Chulaevsky.  On resonances in disordered multi-particle systems , 2012 .

[84]  E. Scoppola,et al.  Localization inv-dimensional incommensurate structures , 1983 .

[85]  D. Damanik,et al.  Absolute continuity of the integrated density of states for the almost Mathieu operator with non-critical coupling , 2007, 0711.4291.

[86]  A. Aronov,et al.  Effects of electron-electron collisions with small energy transfers on quantum localisation , 1982 .

[87]  R. Carmona.  One-dimensional Schrödinger operators with random potentials: A survey , 1985 .

[88]  A. Klein,et al.  Localization for random Schrödinger operators with correlated potentials , 1991 .

[89]  A. Avila,et al.  The Ten Martini Problem , 2009 .

[90]  V. Chulaevsky.  Fixed-energy multi-particle MSA implies dynamical localization , 2012, 1206.1952.

[91]  Peter Stollmann,et al.  Caught by disorder , 2001 .

[92]  Dirk Hundertmark,et al.  A short introduction to Anderson localization , 2007 .

[93]  B. Simon,et al.  Almost periodic Schrödinger operators II. The integrated density of states , 1983 .

[94]  Werner Kirsch,et al.  An Invitation to Random Schr¨ odinger operators , 2007 .

[95]  Joaquim Puig.  Cantor Spectrum for the Almost Mathieu Operator , 2004 .

[96]  Y. Suhov,et al.  Eigenfunctions in a Two-Particle Anderson Tight Binding Model , 2008, 0810.2190.

[97]  L. Pastur,et al.  Introduction to the Theory of Disordered Systems , 1988 .

[98]  Interacting electrons in disordered wires: Anderson localization and low-T transport. , 2005, Physical review letters.

[99]  Y. Sinai.  Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential , 1987 .

[100]  David Ruelle,et al.  A remark on bound states in potential-scattering theory , 1969 .

[101]  L. Thomas,et al.  Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators , 1973 .

[102]  A. Klein,et al.  A comprehensive proof of localization for continuous Anderson models with singular random potentials , 2011, 1105.0213.

[103]  D. Grempel,et al.  Localization in a d-dimensional incommensurate structure , 1984 .

[104]  D. Basko,et al.  On the problem of many-body localization , 2006, cond-mat/0602510.

[105]  A. Avila,et al.  Hölder Continuity of Absolutely Continuous Spectral Measures for One-Frequency Schrödinger Operators , 2009, 0912.3246.

[106]  B. Simon,et al.  Almost periodic Schrödinger operators , 1981 .

[107]  L. Pastur,et al.  An exactly solvable model of a multidimensional incommensurate structure , 1984 .

[108]  R. Bolstein,et al.  Expansions in eigenfunctions of selfadjoint operators , 1968 .

[109]  J. Bourgain,et al.  On localization in the continuous Anderson-Bernoulli model in higher dimension , 2005 .

[110]  V. Chulaevsky.  Direct Scaling Analysis of localization in disordered systems. II. Multi-particle lattice systems , 2011, 1106.2234.

[111]  M. Aizenman.  LOCALIZATION AT WEAK DISORDER: SOME ELEMENTARY BOUNDS , 1994 .

[112]  Mark Kelbert,et al.  Probability and Statistics by Example , 2014 .

[113]  Konstantin Pankrashkin.  Quasiperiodic surface Maryland models on quantum graphs , 2008, 0811.3308.

[114]  E. Dinaburg,et al.  Methods of KAM-theory for long-range quasi-periodic operators on ℤv. Pure point spectrum , 1993 .

[115]  A. B. D. Monvel,et al.  Wegner-type bounds for a two-particle Anderson model in a continuous space , 2008, 0812.2627.

[116]  D. Shepelyansky,et al.  Coherent propagation of two interacting particles in a random potential. , 1994, Physical review letters.

[117]  L. Pastur,et al.  A pure point spectrum of the stochastic one-dimensional schrödinger operator , 1977 .

[118]  Peter Kuchment,et al.  Analysis on graphs and its applications , 2008 .

[119]  V. Chulaevsky.  A remark on charge transfer processes in multi-particle systems , 2010, 1005.3387.

[120]  I. Herbst,et al.  Green’s functions for the Schrödinger equation with short-range potentials , 1989 .

[121]  Жан Бургейн,et al.  Недавний прогресс в квазипериодических решеточных операторах Шрeдингера и гамильтоновых дифференциальных уравнениях с частными производными@@@Recent progress on quasi-periodic lattice Schrödinger operators and Hamiltonian PDEs , 2004 .

[122]  Y. Suhov,et al.  Anderson localisation for an interacting two-particle quantum system on ${\mathbb Z}$ , 2007, 0705.0657.

[123]  P. Anderson,et al.  A selfconsistent theory of localization , 1973 .

[124]  Y. Sinai,et al.  The one-dimensional Schrödinger equation with a quasiperiodic potential , 1975 .

[125]  E. Abrahams.  50 Years of Anderson Localization , 2010 .

[126]  F. Martinelli,et al.  On absence of diffusion near the bottom of the spectrum for a random Schrödinger operator onL2(ℝ)+ , 1984 .

[127]  M. Aizenman,et al.  Localization at large disorder and at extreme energies: An elementary derivations , 1993 .

[128]  L. Eliasson.  Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum , 1997 .

[129]  S A Molčanov,et al.  THE STRUCTURE OF EIGENFUNCTIONS OF ONE-DIMENSIONAL UNORDERED STRUCTURES , 1978 .

[130]  F. Klopp,et al.  The Integrated Density of States for an Interacting Multiparticle Homogeneous Model and Applications to the Anderson Model , 2009 .

[131]  A. P. Shostak.  E-compact extensions of topological spaces , 1974 .

[132]  M. Aizenman,et al.  Communications in Mathematical Physics Finite-Volume Fractional-Moment Criteria for Anderson Localization , 2001 .

[133]  P. Deift,et al.  The Absolutely Continuous Spectrum in One Dimension , 1983 .

[134]  R. Carmona,et al.  Spectral Theory of Random Schrödinger Operators , 1990 .

[135]  Nevill Mott,et al.  Metal-Insulator Transition , 1968 .

[136]  Alexander Figotin,et al.  Spectra of Random and Almost-Periodic Operators , 1991 .

[137]  A. Klein.  Multiscale Analysis and Localization of Random Operators , 2007, 0708.2292.

[138]  J. Bourgain.  Recent progress on quasi-periodic lattice Schrödinger operators and Hamiltonian PDEs , 2004 .

[139]  W. Amrein,et al.  Characterization of bound states and scattering states in quantum mechanics , 1973 .

[140]  G. M. Graf,et al.  A Remark on the Estimate of a Determinant by Minami , 2007 .

[141]  M. Aizenman,et al.  Complete Dynamical Localization in Disordered Quantum Multi-Particle Systems , 2009, 0909.5432.

[142]  T. Spencer,et al.  Positive Lyapunov exponents for a class of deterministic potentials , 1995 .

[143]  A. Joye,et al.  Mathematical Physics of Quantum Mechanics , 2005, math-ph/0506056.

[144]  A. B. D. Monvel,et al.  Wegner-type Bounds for a Multi-particle Continuous Anderson Model with an Alloy-type External Potential , 2008, 0812.2621.

[145]  D. Wiersma,et al.  Fifty years of Anderson localization , 2009 .

[146]  Yvan Velenik,et al.  Ballistic phase of self-interacting random walks , 2007, 0710.3095.

[147]  W. Kirsch.  A Wegner estimate for multi-particle random Hamiltonians , 2007, 0704.2664.