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The integrated density of states for random Schrödinger operators
"... We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger 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 i ..."
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Cited by 43 (4 self)
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We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger 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
INTEGRATED DENSITY OF STATES AND WEGNER ESTIMATES FOR RANDOM SCHRÖDINGER OPERATORS
, 2003
"... We survey recent results on spectral properties of random Schrödinger operators. The focus is set on the integrated density of states (IDS). First we present a proof of the existence of a selfaveraging IDS which is general enough to be applicable to random Schrödinger and LaplaceBeltrami operators ..."
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Cited by 14 (4 self)
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We survey recent results on spectral properties of random Schrödinger operators. The focus is set on the integrated density of states (IDS). First we present a proof of the existence of a selfaveraging IDS which is general enough to be applicable to random Schrödinger and LaplaceBeltrami operators on manifolds. Subsequently we study more specific models in Euclidean space, namely of alloy type, and concentrate on the regularity properties of the IDS. We discuss the role of the integrated density of states and its regularity properties for the spectral analysis of random Schrödinger operators, particularly in relation to localisation. Proofs of the central results are given in detail. Whenever there are alternative proofs, the different approaches are compared.
Series Expansions of Lyapunov Exponents and Forgetful Monoids
, 2000
"... We consider Lyapunov exponents of random iterates of monotone homogeneous maps. We assume that the images of some iterates are lines, with positive probability. Using this memoryloss property which holds generically for random products of matrices over the maxplus semiring, and in particular, for ..."
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Cited by 5 (0 self)
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We consider Lyapunov exponents of random iterates of monotone homogeneous maps. We assume that the images of some iterates are lines, with positive probability. Using this memoryloss property which holds generically for random products of matrices over the maxplus semiring, and in particular, for Tetrislike heaps of pieces models, we give a series expansion formula for the Lyapunov exponent, as a function of the probability law. In the case of rational probability laws, we show that the Lyapunov exponent is an analytic function of the parameters of the law, in a domain that contains the absolute convergence domain of a partition function associated to a special "forgetful" monoid, defined by generators and relations.
SCHRÖDINGER OPERATORS WITH DYNAMICALLY DEFINED POTENTIALS: A SURVEY
, 2014
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Series Expansions of Lyapunov Exponents and
, 2000
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
PROPERTIES OF THE COLUMNS IN THE INFINITE PRODUCTS OF NONNEGATIVE MATRICES
, 2013
"... Abstract. Given a sequence (An)n∈N of complexvalued d × d matrices, the normalized product matrix A1...An ‖A1...An‖ in general diverges, so we are rather interested by the convergence of the normalized columns of A1... An. Now we assume that the An are nonnegative. Then in most cases the dominant c ..."
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Abstract. Given a sequence (An)n∈N of complexvalued d × d matrices, the normalized product matrix A1...An ‖A1...An‖ in general diverges, so we are rather interested by the convergence of the normalized columns of A1... An. Now we assume that the An are nonnegative. Then in most cases the dominant columns of A1... An have the same limit in direction when n → ∞. In the main theorem we give a sufficient condition for this limit to exist. The other columns may have distinct limits but are negligible in norm so, A1...AnV ‖A1...AnV ‖ if V is a positive ddimensional column vector, the sequence of column vectors converges. The second part of the paper concerns the measures defined by Bernoulli convolution. In certain cases the values of such a measure can be computed by means of products of the form A1... AnV, where An belongs to a finite set of matrices and V is A1...AnV ‖A1...AnV ‖ a positive column vector, and if converges uniformly we can deduce the local properties of the measure. But in general the matrices are great, have much zero entries, and it is not obvious that they satisfy the condition of the theorem. So we choose one example where the matrices have a reasonable size, and we explain how to proceed to apply the theorem and obtain the Gibbs and multifractal properties of the measure.