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213
Modified Prüfer and EFGP Transforms and the Spectral Analysis of OneDimensional Schrödinger Operators
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1997
"... Using control of the growth of the transfer matrices, we discuss the spectral analysis of continuum and discrete halfline Schrödinger operators with slowly decaying potentials. Among our results we show if V (x) = ∑∞ n=1 anW (x − xn), where W has compact support and xn/xn+1 → 0, then H has purely ..."
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Cited by 66 (21 self)
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Using control of the growth of the transfer matrices, we discuss the spectral analysis of continuum and discrete halfline Schrödinger operators with slowly decaying potentials. Among our results we show if V (x) = ∑∞ n=1 anW (x − xn), where W has compact support and xn/xn+1 → 0, then H has purely a.c. (resp. purely s.c.) spectrum on (0, ∞) if ∑ a2 n <∞(resp. ∑ a2 n = ∞). For λn−1/2an potentials, where an are independent, identically distributed random variables with E(an) =0,E(a2 n)=1,and λ < 2, we find singular continuous spectrum with explicitly computable fractional Hausdorff dimension.
Internal Lifshits Tails For Random Perturbations Of Periodic Schrödinger Operators
"... . Let H be a \Gammaperiodic Schrodinger operator acting on L 2 (R d ) and consider the random Schrodinger operator H! = H + V! where V! (x) = X fl2\Gamma ! fl V (x \Gamma fl) (here V is a positive potential and (! fl ) fl2\Gamma a collection of positive i.i.d random variables). We prove tha ..."
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Cited by 42 (7 self)
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. Let H be a \Gammaperiodic Schrodinger operator acting on L 2 (R d ) and consider the random Schrodinger operator H! = H + V! where V! (x) = X fl2\Gamma ! fl V (x \Gamma fl) (here V is a positive potential and (! fl ) fl2\Gamma a collection of positive i.i.d random variables). We prove that, at the edge of a gap of H that is not filled in for H! , the integrated density of states of H! has a Lifshits tail behaviour if and only if the integrated density of states of H is nondegenerate. R' esum' e. Soient H un op'erateur de Schrodinger \Gammap'eriodique agissant sur L 2 (R d ), V un potentiel positif et (! fl ) fl2Z d une famille de variables al'eatoires i.i.d positives. Consid'erons l'op'erateur de Schrodinger al'eatoire H! = H + V! o`u V! (x) = X fl2Z d ! fl V (x \Gamma fl). On montre que, au bord d'une lacune spectrale de H qui n'est pas combl'e pour H! la densit'e d"etats int'egr'ee de H! a un comportement asymptotique de Lifshits si et seulement si la densit'e d"...
A characterization of the Anderson metalinsulator transport transition
 Duke Math. J
"... We investigate the Anderson metalinsulator transition for random Schrödinger operators. We define the strong... ..."
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Cited by 41 (17 self)
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We investigate the Anderson metalinsulator transition for random Schrödinger operators. We define the strong...
Moment Analysis for Localization in Random Schrödinger Operators
, 2005
"... We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the ..."
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Cited by 38 (12 self)
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We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonancediffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the LifshitzKrein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weakL¹ estimate concerning the boundaryvalue distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.
Localization of Classical Waves I: Acoustic Waves.
 Commun. Math. Phys
, 1996
"... We consider classical acoustic waves in a medium described by a position dependent mass density %(x). We assume that %(x) is a random perturbation of a periodic function % 0 (x) and that the periodic acoustic operator A 0 = \Gammar \Delta 1 %0 (x) r has a gap in the spectrum. We prove the existe ..."
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Cited by 37 (0 self)
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We consider classical acoustic waves in a medium described by a position dependent mass density %(x). We assume that %(x) is a random perturbation of a periodic function % 0 (x) and that the periodic acoustic operator A 0 = \Gammar \Delta 1 %0 (x) r has a gap in the spectrum. We prove the existence of localized waves, i.e., finite energy solutions of the acoustic equations with the property that almost all of the wave's energy remains in a fixed bounded region of space at all times, with probability one. Localization of acoustic waves is a consequence of Anderson localization for the selfadjoint operators A = \Gammar \Delta 1 %(x) r on L 2 (R d ). We prove that, in the random medium described by %(x), the random operator A exhibits Anderson localization inside the gap in the spectrum of A 0 . This is shown even in situations when the gap is totally filled by the spectrum of the random operator; we can prescribe random environments that ensure localization in almost the wh...
FiniteVolume FractionalMoment Criteria for Anderson Localization
, 2000
"... A technically convenient signature of localization, exhibited by discrete operators with random potentials, is exponential decay of the fractional moments of the Green function within the appropriate energy ranges. Known implications include: spectral localization, absence of level repulsion, strong ..."
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Cited by 33 (4 self)
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A technically convenient signature of localization, exhibited by discrete operators with random potentials, is exponential decay of the fractional moments of the Green function within the appropriate energy ranges. Known implications include: spectral localization, absence of level repulsion, strong form of dynamical localization, and a related condition which plays a significant role in the quantization of the Hall conductance in twodimensional Fermi gases. We present a family of finitevolume criteria which, under some mild restrictions on the distribution of the potential, cover the regime where the fractional moment decay condition holds. The constructive criteria permit to establish this condition at spectral band edges, provided there are sufficient `Lifshitz tail estimates' on the density of states. They are also used here to conclude that the fractional moment condition, and thus the other manifestations of localization, are valid throughout the regime covered by the "multisca...
Spectral properties of the laplacian on bondpercolation graphs
"... Abstract Bondpercolation graphs are random subgraphs of the ddimensional integer lattice generated by a standard bondpercolation process. The associated graph Laplacians, subject to Dirichlet or Neumann conditions at cluster boundaries, represent bounded, selfadjoint, ergodic random operators wi ..."
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Cited by 33 (9 self)
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Abstract Bondpercolation graphs are random subgraphs of the ddimensional integer lattice generated by a standard bondpercolation process. The associated graph Laplacians, subject to Dirichlet or Neumann conditions at cluster boundaries, represent bounded, selfadjoint, ergodic random operators with offdiagonal disorder. They possess almost surely the nonrandom spectrum [0, 4d] and a selfaveraging integrated density of states. The integrated density of states is shown to exhibit Lifshits tails at both spectral edges in the nonpercolating phase. While the characteristic exponent of the Lifshits tail for the Dirichlet (Neumann) Laplacian at the lower (upper) spectral edge equals d/2, and thus depends on the spatial dimension, this is not the case at the upper (lower) spectral edge, where the exponent equals 1/2.
Sum rules and the Szegő condition for orthogonal polynomials on the real line
, 2002
"... We study the Case sum rules, especially C 0, for general Jacobi matrices. We establish situations where the sum rule is valid. Applications include an extension of Shohat's theorem to cases with an infinite point spectrum and a proof that if lim n(an 1) = and lim nbn = exist and 2 jj, then th ..."
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Cited by 32 (17 self)
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We study the Case sum rules, especially C 0, for general Jacobi matrices. We establish situations where the sum rule is valid. Applications include an extension of Shohat's theorem to cases with an infinite point spectrum and a proof that if lim n(an 1) = and lim nbn = exist and 2 jj, then the Szegő condition fails.
Absolutely continuous spectra of quantum tree graphs with weak disorder
"... Abstract: We consider the Laplacian on a rooted metric tree graph with branching number K ≥ 2 and random edge lengths given by independent and identically distributed bounded variables. Our main result is the stability of the absolutely continuous spectrum for weak disorder. A useful tool in the dis ..."
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Cited by 29 (5 self)
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Abstract: We consider the Laplacian on a rooted metric tree graph with branching number K ≥ 2 and random edge lengths given by independent and identically distributed bounded variables. Our main result is the stability of the absolutely continuous spectrum for weak disorder. A useful tool in the discussion is a function which expresses a directional transmission amplitude to infinity and forms a generalization of the WeylTitchmarsh function to trees. The proof of the main result rests on upper bounds on the range of fluctuations of this quantity in the limit of weak disorder. Contents 1. Introduction.................................
Twoparameter spectral averaging and localization for nonmonotonic random Schrödinger operators
 TRANS. AMER. MATH. SOC
, 2001
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