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30
An invitation to random Schrödinger operators
, 2007
"... This review is an extended version of my mini course at the États de la recherche: Opérateurs de Schrödinger aléatoires at the Université Paris 13 in June 2002, a summer school organized by Frédéric Klopp. These lecture notes try to give some of the basics of random Schrödinger operators. They are m ..."
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Cited by 50 (7 self)
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This review is an extended version of my mini course at the États de la recherche: Opérateurs de Schrödinger aléatoires at the Université Paris 13 in June 2002, a summer school organized by Frédéric Klopp. These lecture notes try to give some of the basics of random Schrödinger operators. They are meant for nonspecialists and require only minor previous knowledge about functional analysis and probability theory. Nevertheless this survey includes complete proofs of Lifshitz tails and Anderson localization. Copyright by the author. Copying for academic purposes is permitted.
Quantum diffusion of the random Schrödinger evolution in the scaling limit II. The recollision diagrams
"... We consider random Schrödinger equations on R d for d ≥ 3 with a homogeneous AndersonPoisson type random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. The space and time variables scale as x ∼ λ −2−κ/2,t ∼ λ −2−κ with 0 < κ < κ0(d). We prove that, in t ..."
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Cited by 21 (5 self)
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We consider random Schrödinger equations on R d for d ≥ 3 with a homogeneous AndersonPoisson type random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. The space and time variables scale as x ∼ λ −2−κ/2,t ∼ λ −2−κ with 0 < κ < κ0(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψt converges weakly to the solution of a heat equation in the space variable x for arbitrary L 2 initial data. The proof is based on analyzing the phase cancellations of multiple scatterings on the random potential by expanding the propagator into a sum of Feynman graphs. In this paper we consider the nonrecollision graphs and prove that the amplitude of the nonladder diagrams is smaller than their “naive size ” by an extra λ c factor per non(anti)ladder vertex for some c> 0. This is the first rigorous result showing that the
Anderson localization for radial treelike random quantum graphs
, 2008
"... We prove that certain random models associated with radial, treelike, rooted quantum graphs exhibit Anderson localization at all energies. The two main examples are the random length model (RLM) and the random Kirchhoff model (RKM). In the RLM, the lengths of each generation of edges form a family ..."
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Cited by 10 (0 self)
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We prove that certain random models associated with radial, treelike, rooted quantum graphs exhibit Anderson localization at all energies. The two main examples are the random length model (RLM) and the random Kirchhoff model (RKM). In the RLM, the lengths of each generation of edges form a family of independent, identically distributed random variables (iid). For the RKM, the iid random variables are associated with each generation of vertices and moderate the current flow through the vertex. We consider extensions to various families of decorated graphs and prove stability of localization with respect to decoration. In particular, we prove Anderson localization for the random necklace model.
Sch’nol’s theorem for strongly local forms
, 2009
"... We prove a variant of Sch’nol’s theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with δ or Kirchhoff boundary conditions. ..."
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Cited by 10 (6 self)
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We prove a variant of Sch’nol’s theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with δ or Kirchhoff boundary conditions.
Stollmann: Eigenfunction expansion for Schrödinger operators on metric graphs (Preprint arXiv:0801.1376
"... Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices. ..."
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Cited by 8 (5 self)
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Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.
Localization on quantum graphs with random vertex couplings
 J. Statist. Phys
"... Abstract. We consider Schrödinger operators on a class of periodic quantum graphs with randomly distributed Kirchhoff coupling constants at all vertices. Using the technique of selfadjoint extensions we obtain conditions for localization on quantum graphs in terms of finite volume criteria for some ..."
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Cited by 7 (0 self)
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Abstract. We consider Schrödinger operators on a class of periodic quantum graphs with randomly distributed Kirchhoff coupling constants at all vertices. Using the technique of selfadjoint extensions we obtain conditions for localization on quantum graphs in terms of finite volume criteria for some energydependent discrete Hamiltonians. These conditions hold in the strong disorder limit and at the spectral edges.
Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model
, 2010
"... We consider Hermitian and symmetric random band matrices H in d � 1 dimensions. The matrix elements Hxy, indexed by x, y ∈ Λ ⊂ Z d, are independent, uniformly distributed random variables if x − y  is less than the band width W, and zero otherwise. We prove that the time evolution of a quantum par ..."
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Cited by 5 (1 self)
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We consider Hermitian and symmetric random band matrices H in d � 1 dimensions. The matrix elements Hxy, indexed by x, y ∈ Λ ⊂ Z d, are independent, uniformly distributed random variables if x − y  is less than the band width W, and zero otherwise. We prove that the time evolution of a quantum particle subject to the Hamiltonian H is diffusive on time scales t ≪ W d/3. We also show that the localization length of the eigenvectors of H is larger than a factor W d/6 times the band width. All results are uniform in the size Λ  of the matrix. AMS Subject Classification: 15B52, 82B44, 82C44
Optimal Wegner estimates for random Schrödinger operators on metric graphs
 in [EKK+ 08] (2008), 409–422
"... Abstract. We consider Schrödinger operators with a random potential of alloy type on infinite metric graphs which obey certain uniformity conditions. For single site potentials of fixed sign we prove that the random Schrödinger operator restricted to a finite volume subgraph obeys a Wegner estimate ..."
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Abstract. We consider Schrödinger operators with a random potential of alloy type on infinite metric graphs which obey certain uniformity conditions. For single site potentials of fixed sign we prove that the random Schrödinger operator restricted to a finite volume subgraph obeys a Wegner estimate which is linear in the volume and reproduces the modulus of continuity of the single site distribution. This improves and unifies earlier results for alloy type models on metric graphs. We discuss applications of Wegner estimates to bounds on the modulus of continuity for the integrated density of states of ergodic Schrödinger operators, as well as to the proof of Anderson localisation via the multiscale analysis. 1.
Persistence under weak disorder of AC spectra of quasiperiodic Schrödinger operators on trees graphs
 Mosc. Math. J
"... Dedicated to Ya. Sinai on the occasion of his seventieth birthday Abstract: We consider radial tree extensions of onedimensional quasiperiodic Schrödinger operators and establish the stability of their absolutely continuous spectra under weak but extensive perturbations by a random potential. The ..."
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Dedicated to Ya. Sinai on the occasion of his seventieth birthday Abstract: We consider radial tree extensions of onedimensional quasiperiodic Schrödinger operators and establish the stability of their absolutely continuous spectra under weak but extensive perturbations by a random potential. The sufficiency criterion for that is the existence of BlochFloquet states for the one dimensional operator corresponding to the radial problem. Key words. Random operators, absolutely continuous spectrum, quasiperiodic cocycles,
Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms
, 2009
"... We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples. ..."
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Cited by 4 (4 self)
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We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples.