The Integrated Density of States for Random Schroedinger Operators

We survey some aspects of the theory of the integrated density of states (IDS) of random Schroedinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is discussed in the second section. One central topic of this survey is the asymptotic behavior of the integrated density of states at the boundary of the spectrum. In particular, we are interested in Lifshitz tails and the occurrence of a classical and a quantum regime. In the last section we discuss regularity properties of the IDS. Our emphasis is on the discussion of fundamental problems and central ideas to handle them. Finally, we discuss further developments and problems of current research.

[1]  J. Gärtner,et al.  Moment asymptotics for the continuous parabolic Anderson model , 2000 .

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

[3]  Ivan Veselić,et al.  Spectral analysis of percolation Hamiltonians , 2004, math-ph/0405006.

[4]  F. Klopp.  Precise High Energy Asymptotics for the Integrated Density of States of an Unbounded Random Jacobi Matrix , 2000 .

[5]  M. Fukushima.  ON THE SPECTRAL DISTRIBUTION OF A DISORDERED SYSTEM AND THE RANGE OF A RANDOM WALK , 1974 .

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

[7]  J. Combes,et al.  Global Continuity of the Integrated Density of States for Random Landau Hamiltonians , 2004 .

[8]  J. Fröhlich,et al.  Analyticity of the density of states and replica method for random schrödinger operators on a lattice , 1984 .

[9]  Peter Muller,et al.  A Survey of Rigorous Results on Random Schrödinger Operators for Amorphous Solids , 2005 .

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

[11]  Werner Kirsch,et al.  On the spectrum of Schrödinger operators with a random potential , 1982 .

[12]  Barry Simon,et al.  Log hölder continuity of the integrated density of states for stochastic Jacobi matrices , 1983 .

[13]  Classical magnetic Lifshits tails in three space dimensions: impurity potentials with slow anisotropic decay , 2002, math-ph/0212078.

[14]  A. Klein,et al.  Smoothness of the density of states in the Anderson model on a one-dimensional strip , 1988 .

[15]  P. Stollmann.  Lifshitz Asymptotics via Linear Coupling of Disorder , 1999 .

[16]  M. Reed,et al.  Methods of Modern Mathematical Physics. 2. Fourier Analysis, Self-adjointness , 1975 .

[17]  G. Stolz,et al.  Localization for random perturbations of periodic Schrödinger operators , 1998 .

[18]  Michael Goldstein,et al.  On the Formation of Gaps in the Spectrum of Schrödinger Operators with Quasi-Periodic Potentials , 2006 .

[19]  Localization for Random Perturbations of Periodic Schrödinger Operators with Regular Floquet Eigenvalues , 2002, math-ph/0510063.

[20]  S. Nakamura.  A remark on the Lifshitz tail for Schrödinger operator with random magnetic field , 2002 .

[21]  B. Simon.  SCHRODINGER OPERATORS IN THE TWENTY-FIRST CENTURY , 1986 .

[22]  J. Kingman.  Subadditive Ergodic Theory , 1973 .

[23]  B. Simon.  Lifschitz tails for the Anderson model , 1985 .

[24]  EXISTENCE AND UNIQUENESS OF THE INTEGRATED DENSITY OF STATES FOR SCHRÖDINGER OPERATORS WITH MAGNETIC FIELDS AND UNBOUNDED RANDOM POTENTIALS , 2000, math-ph/0010013.

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

[26]  The Band-Edge Behavior of the Density of Surfacic States , 2004, math-ph/0407051.

[27]  M. Meerschaert.  Regular Variation in R k , 1988 .

[28]  Spectral properties of the Laplacian on bond-percolation graphs , 2004, math-ph/0407047.

[29]  W. Kirsch,et al.  Lifshitz-Tails and Non-Lifshitz-Tails for One-Dimensional Random Point Interactions , 1990 .

[30]  M. Barlow.  Random walks on supercritical percolation clusters , 2003, math/0302004.

[31]  P. Stollmann,et al.  Discontinuities of the Integrated Density of States for Random Operators on Delone Sets , 2002, math-ph/0208027.

[32]  A. Sznitman,et al.  Some connections between excursion theory and the discrete Schrödinger equation with random potentials , 1987 .

[33]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

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

[35]  B. Simon.  Spectral averaging and the Krein spectral shift , 1998 .

[36]  W. Müller.  The spectral shift function , 1987 .

[37]  The Lp-Theory of the Spectral Shift Function,¶the Wegner Estimate, and the Integrated Density¶of States for Some Random Operators , 2001 .

[38]  L. Pastur,et al.  ON THE SPECTRUM OF THE ONE-DIMENSIONAL SCHRÖDINGER EQUATION WITH A RANDOM POTENTIAL , 1970 .

[39]  R. Metzger.  Assymptotische Eigenschaften im Wechselspiel von Diffusion und Wellenausbreitung in zufälligen Medien , 2005 .

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

[41]  Long-time tails in the parabolic Anderson model , 2000, math-ph/0004014.

[42]  J. Combes,et al.  Hölder continuity of the integrated density of states for some random operators at all energies , 2003 .

[43]  S. Warzel,et al.  Poissonian Obstacles with Gaussian Walls Discriminate Between Classical and Quantum Lifshits Tailing in Magnetic Fields , 1999 .

[44]  A. Sznitman.  Lifschitz tail and Wiener sausage, II , 1990 .

[45]  H. Nagai,et al.  On an Asymptotic Property of Spectra of a Random Difference Operator , 1975 .

[46]  A. Klein,et al.  A supersymmetric transfer matrix and differentiability of the density of states in the one-dimensional Anderson model , 1986 .

[47]  The fate of lifshits tails in magnetic fields , 1995, cond-mat/9503057.

[48]  Fine Properties of the Integrated Density of States and a Quantitative Separation Property of the Dirichlet Eigenvalues , 2005, math-ph/0501005.

[49]  F. Martinelli,et al.  On the ergodic properties of the specrum of general random operators. , 1982 .

[50]  Peter Stollmann,et al.  Caught by Disorder: Bound States in Random Media , 2001 .

[51]  W. Kirsch.  Wegner estimates and Anderson localization for alloy-type potentials , 1996 .

[52]  W. Kirsch,et al.  Wegner estimate for sparse and other generalized alloy type potentials , 2002 .

[53]  W. Fischer,et al.  Existence of the Density of States for Multi-Dimensional¶Continuum Schrödinger Operators with¶Gaussian Random Potentials , 1997 .

[54]  F. Klopp.  Lifshitz tails for random perturbations of periodic Schrödinger operators , 2002 .

[55]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[56]  D. Thouless,et al.  Electrons in disordered systems and the theory of localization , 1974 .

[57]  H. Englisch,et al.  Random Hamiltonians ergodic in all but one direction , 1990 .

[58]  S. Agmon.  On Positive Solutions of Elliptic Equations with Periodic Coefficients in N, Spectral Results and Extensions to Elliptic Operators on Riemannian Manifolds , 1984 .

[59]  D. Yafaev.  Mathematical scattering theory , 1992 .

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

[61]  F. Martinelli,et al.  A rigorous replica trick approach to Anderson localization in one dimension , 1986 .

[62]  Ayham Chahrour.  On the Spectrum of the Schrödinger Operator with Periodic Surface Potential , 2000 .

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

[64]  A. Sznitman.  Lifschitz Tail and Wiener Sausage on hyperbolic space , 1989 .

[65]  W. Kirsch,et al.  The effect of boundary conditions on the density of states for random Schrödinger operators , 1986 .

[66]  S. Nakamura.  Lifshitz Tail for Schrödinger Operator¶with Random Magnetic Field , 2000 .

[67]  F. Martinelli,et al.  On the large-coupling-constant behavior of the Liapunov exponent in a binary alloy , 1987 .

[68]  Existence of the density of states for one-dimensional alloy-type potentials with small support , 2002, math-ph/0204030.

[69]  W. Wreszinski,et al.  On the Lifschitz singularity and the tailing in the density of states for random lattice systems , 1979 .

[70]  L. Pastur.  On the distribution of the eigenvalues of the Schrödinger equation with a random potential , 1972 .

[71]  B. Halperin.  Properties of a Particle in a One‐Dimensional Random Potential , 2007 .

[72]  R. Schrader,et al.  Regularity of the Surface Density of States , 2001 .

[73]  M. Reed.  Methods of Modern Mathematical Physics. I: Functional Analysis , 1972 .

[74]  Hans L. Cycon,et al.  Schrodinger Operators: With Application to Quantum Mechanics and Global Geometry , 1987 .

[75]  Peter Muller,et al.  Lifshitz tails for spectra of Erdős–Rényi random graphs , 2005, math-ph/0502054.

[76]  F. Delyon,et al.  Remark on the continuity of the density of states of ergodic finite difference operators , 1984 .

[77]  D. Newton,et al.  ERGODIC THEOREMS (de Gruyter Studies in Mathematics 6) , 1986 .

[78]  B. Simon,et al.  An optimalLp-bound on the Krein spectral shift function , 2002 .

[79]  L. Pastur.  Spectra of Random Self Adjoint Operators , 1973 .

[80]  Srinivasa Varadhan,et al.  Asymptotics for the wiener sausage , 1975 .

[81]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[82]  L. Pastur.  Spectral properties of disordered systems in the one-body approximation , 1980 .

[83]  P. Hislop,et al.  The Wegner estimate and the integrated density of states for some random operators , 2002 .

[84]  Werner Kirsch,et al.  On the density of states of Schrodinger operators with a random potential , 1982 .

[85]  J. Combes,et al.  Landau Hamiltonians with Unbounded Random Potentials , 1997 .

[86]  S. Nakamura.  Lifshitz Tail for 2D Discrete Schrödinger Operator with Random Magnetic Field , 2000 .

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

[88]  F. Klopp,et al.  Lifshitz tails for 2-dimensional random Schrödinger operators , 2002 .

[89]  W. Kirsch.  Small perturbations and the eigenvalues of the Laplacian on large bounded domains , 1987 .

[90]  J. Lamperti.  Stochastic processes : a survey of the mathematical theory , 1979 .

[91]  Barry Simon,et al.  Subharmonicity of the Lyaponov index , 1983 .

[92]  B. Simon.  Internal Lifschitz tails , 1987 .

[93]  F. Klopp,et al.  Internal Lifshitz tails for random Schr?dinger operators , 2000 .

[94]  B. Simon,et al.  Singular continuous spectrum for a class of almost periodic Jacobi matrices , 1982 .

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

[96]  The Absolute Continuity of the Integrated Density¶of States for Magnetic Schrödinger Operators¶with Certain Unbounded Random Potentials , 2001, math-ph/0105046.

[97]  Steffen Klassert.  Unschärfe, der supersymmetrische Replica Trick und Lokalisierung , 2006 .

[98]  S. Nakao.  On the spectral distribution of the Schrödinger operator with random potential , 1977 .

[99]  A. Sznitman.  Lifschitz tail on hyperbolic space: Neumann conditions , 1990 .

[100]  I. Lifshitz,et al.  Reviews of Topical Problems: Energy Spectrum Structure and Quantum States of Disordered Condensed Systems , 1965 .

[101]  THE DENSITY OF STATES AND THE SPECTRAL SHIFT DENSITY OF RANDOM SCHRÖDINGER OPERATORS , 2000, math-ph/0011033.

[102]  L. Pastur,et al.  Lifshitz Tails for Random Schrödinger Operators¶with Negative Singular Poisson Potential , 1999 .

[103]  On spectra of random Schrödinger operators with magnetic fields , 1994 .

[104]  Bounds on the Spectral Shift Function and the Density of States , 2004, math-ph/0412078.

[105]  J. Combes,et al.  Spectral averaging, perturbation of singular spectra, and localization , 1996 .

[106]  B. Simon,et al.  Lifshitz tails for periodic plus random potentials , 1986 .

[107]  F. Klopp,et al.  Endpoints of the spectrum of periodic operators are generically simple , 2000 .

[108]  A. Klein,et al.  Anomalies in the one-dimensional Anderson model at weak disorder , 1990 .

[109]  A. Klein,et al.  Regularity of the invariant measure and of the density of states in the one-dimensional Anderson model , 1990 .

[110]  P. Stollmann,et al.  Lifshitz Asymptotics and Localization for Random Quantum Waveguides , 2000 .

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

[112]  U. Krengel,et al.  Ergodic theorems for superadditive processes. , 1981 .

[113]  D. Damanik,et al.  LOCALIZATION FOR ONE DIMENSIONAL, CONTINUUM, BERNOULLI-ANDERSON MODELS , 2000, math-ph/0010016.

[114]  F. Klopp.  Weak Disorder Localization and Lifshitz Tails: Continuous Hamiltonians , 2002 .

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

[116]  P. Kuchment.  Floquet Theory for Partial Differential Equations , 1993 .

[117]  S. Varadhan.  Large Deviations and Applications , 1984 .

[118]  B. Simon.  Functional integration and quantum physics , 1979 .

[119]  J. Moser,et al.  The rotation number for almost periodic potentials , 1983 .

[120]  C. Shubin,et al.  Some Harmonic Analysis Questions Suggested by Anderson-Bernoulli Models , 1998 .

[121]  I. Jacobi.  Localization in General One Dimensional Random Systems , 1985 .

[122]  G. Mezincescu.  Internal Lifschitz singularities of disordered finite-difference Schrödinger operators , 1986 .

[123]  J. Combes,et al.  Localization Near Band Edges For Random Schr Odinger Operators , 1997 .

[124]  N. Ueki.  Simple Examples of Lifschitz Tails in Gaussian Random Magnetic Fields , 2000 .

[125]  Barry Simon,et al.  Comparison theorems for the gap of Schrödinger òperators , 1987 .

[126]  F. Martinelli,et al.  Large deviations and Lifshitz singularity of the integrated density of states of random Hamiltonians , 1983 .

[127]  W. Kirsch.  An Invitation to Random Schroedinger operators , 2007, 0709.3707.

[128]  Abel Klein,et al.  Random Schrödinger operators , 2008 .

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

[130]  E. L. Page.  Repartition d'etat d'un operateur de Schrödinger aleatoire Distribution empirique des valeurs propres d'une matrice de Jacobi , 1984 .

[131]  On the Lipschitz continuity of the integrated density of states for sign-indefinite potentials , 2004, math-ph/0408013.

[132]  L. Pastur.  Behavior of some Wiener integrals as t→∞ and the density of states of Schrödinger equations with random potential , 1977 .

[133]  J. Gärtner,et al.  Parabolic problems for the Anderson model , 1998 .

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

[135]  A. Sznitman.  Brownian motion, obstacles, and random media , 1998 .

[136]  H. Najar.  Lifshitz tails for random acoustic operators , 2003 .

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

[138]  Reinhard Lang,et al.  Spectral Theory of Random Schrödinger Operators , 1991 .

[139]  Anderson Localization and Lifshits Tails for Random Surface Potentials , 2004, math-ph/0412079.

[140]  J. Combes,et al.  Localization for Some Continuous, Random Hamiltonians in d-Dimensions , 1994 .

[141]  Wolff,et al.  Some rigorous results for the Anderson model. , 1985, Physical review letters.

[142]  H. McKean,et al.  Hill’s surfaces and their theta functions , 1978 .

[143]  L. Erdős.  Lifschitz tail in a magnetic field: the nonclassical regime , 1998 .

[144]  Barry Simon,et al.  Spectral analysis of rank one perturbations and applications , 1995 .

[145]  B. Helffer,et al.  Asymptotic of the density of states for the Schrödinger operator with periodic electric potential , 1998 .

[146]  G. Mezincescu.  Internal Lifschitz singularities for one dimensional Schrödinger operators , 1993 .

[147]  B. Lautrup,et al.  Products of random matrices. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[148]  M.,et al.  Brownian Motion and Harnack Inequality for Schrodinger Operators , 2022 .

[149]  B. Simon,et al.  Schrödinger Semigroups , 2007 .

[150]  H. Najar.  Asymptotic Behavior of the Integrated Density of States of Acoustic Operators with Random Long Range Perturbations , 2004 .

[151]  D. Hundertmark,et al.  Continuity properties of Schrödinger semigroups with magnetic fields , 1998, math-ph/9808004.

[152]  P. Hislop,et al.  The Integrated Density of States for Some Random Operators with Nonsign Definite Potentials , 2002 .

[153]  P. Deift.  Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach , 2000 .

[154]  B. Simon,et al.  Harmonic analysis on SL(2,R) and smoothness of the density of states in the one-dimensional Anderson model , 1985 .

[155]  J. B. McLeod.  THE SPECTRAL THEORY OF PERIODIC DIFFERENTIAL EQUATIONS , 1975 .

[156]  Peter Muller,et al.  Spectral asymptotics of the Laplacian on supercritical bond-percolation graphs☆ , 2007 .

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

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

[159]  F. Klopp.  Band Edge Behavior of the Integrated Density of States of Random Jacobi Matrices in Dimension 1 , 1998 .

[160]  Lifshits Tails Caused by Anisotropic Decay: The Emergence of a Quantum-Classical Regime , 2003, math-ph/0310033.

[161]  G. Raikov,et al.  Lifshitz Tails in Constant Magnetic Fields , 2005, math-ph/0509022.

[162]  A. Chahrour.  DENSITE INTEGREE D'ETATS SURFACIQUES ET FONCTION GENERALISEE DE DEPLACEMENT SPECTRAL POUR UN OPERATEUR DE SCHRODINGER SURFACIQUE ERGODIQUE , 1999 .

[163]  D. Yafaev.  Mathematical Scattering Theory: General Theory , 1992 .

[164]  Barry Simon,et al.  Methods of modern mathematical physics. III. Scattering theory , 1979 .

[165]  G. Mezincescu.  Lifschitz singularities for periodic operators plus random potentials , 1987 .

[166]  F. Klopp.  INTERNAL LIFSHITS TAILS FOR RANDOM PERTURBATIONS OF PERIODIC SCHRODINGER OPERATORS , 1999 .

[167]  Kirsch,et al.  Density of surface states in discrete models. , 1988, Physical review letters.

[168]  Michael Goldstein,et al.  Holder continuity of the integrated density of states for quasi-periodic Schrodinger equations and averages of shifts of subharmonic functions , 2001 .

[169]  S. Nakamura.  Spectral Shift Function for Trapping Energies¶in the Semiclassical Limit , 1999 .

[170]  F. Klopp.  Localization for some continuous random Schrödinger operators , 1995 .

[171]  D. Herbert,et al.  Localized states in disordered systems , 1971 .

[172]  F. Delyon,et al.  The rotation number for finite difference operators and its properties , 1983 .

[173]  J. Lacroix,et al.  Regularity of the density of states in the Anderson model on a strip for potentials with singular continuous distributions , 1989 .