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LOCALIZATION NEAR FLUCTUATION BOUNDARIES VIA FRACTIONAL MOMENTS AND APPLICATIONS
, 2006
"... We present a new, short, selfcontained proof of localization properties of multidimensional continuum random Schrödinger operators in the fluctuation boundary regime. Our method is based on the recent extension of the fractional moment method to continuum models in [2], but does not require the ra ..."
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Cited by 7 (5 self)
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We present a new, short, selfcontained proof of localization properties of multidimensional continuum random Schrödinger operators in the fluctuation boundary regime. Our method is based on the recent extension of the fractional moment method to continuum models in [2], but does not require the random potential to satisfy a covering condition. Applications to random surface potentials and potentials with random displacements are included.
Surface States and Spectra
, 2000
"... Let Z d+1 + = Z d Z+,letH 0 be the discrete Laplacian on the Hilbert space l 2 (Z d+1 + ) with a Dirichlet boundary condition, and let V be a potential supported on the boundary #Z d+1 + .We introduce the notions of surface states and surface spectrum of the operator H = H 0 + V and ex ..."
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Cited by 6 (1 self)
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Let Z d+1 + = Z d Z+,letH 0 be the discrete Laplacian on the Hilbert space l 2 (Z d+1 + ) with a Dirichlet boundary condition, and let V be a potential supported on the boundary #Z d+1 + .We introduce the notions of surface states and surface spectrum of the operator H = H 0 + V and explore their properties. Our main result is that if the potential V is random and if the disorder is either large or small enough, then in dimension two H has no surface spectrum on #(H 0 ) with probability one. To prove this result we combine AizenmanMolchanov theory with techniques of scattering theory.
Absence of continuous spectral types for certain nonstationary random Schrödinger operators
, 2005
"... We consider continuum random Schrödinger operators of the type Hω = −∆+V0 + Vω with a deterministic background potential V0. We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of − ∆ +V0. The models we treat include random sur ..."
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Cited by 3 (1 self)
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We consider continuum random Schrödinger operators of the type Hω = −∆+V0 + Vω with a deterministic background potential V0. We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of − ∆ +V0. The models we treat include random surface potentials as well as sparse or slowly decaying random potentials. In particular, we establish absence of absolutely continuous surface spectrum for random potentials supported near a onedimensional surface (“random tube”) in arbitrary dimension.
Spectral Theory of Corrugated Surfaces
, 2001
"... We discuss spectral and scattering theory of the discrete Laplacian limited to a halfspace. The interesting properties of such operators stem from the imposed boundary condition and are related to certain phenomena in surface physics. 1. ..."
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Cited by 2 (0 self)
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We discuss spectral and scattering theory of the discrete Laplacian limited to a halfspace. The interesting properties of such operators stem from the imposed boundary condition and are related to certain phenomena in surface physics. 1.
Dynamical localization for continuum random surface models By
"... Abstract. We prove Anderson localization and strong dynamical localization for random surface models in R d. 1. Introduction, the model, and the results. Spectral and scattering theory for mathematical models of rough surfaces has attracted considerable interest in recent years, as witnessed in [2, ..."
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Cited by 2 (0 self)
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Abstract. We prove Anderson localization and strong dynamical localization for random surface models in R d. 1. Introduction, the model, and the results. Spectral and scattering theory for mathematical models of rough surfaces has attracted considerable interest in recent years, as witnessed in [2, 3, 7–13, 16]. One of the reasons is that these models exhibit a metalinsulator transition. This transition is expected for typical random models in dimensions three and higher but, unfortunately, a
On the Spectral and Wave Propagation Properties of the
, 2008
"... We study the discrete Schrödinger operator H in Zd with the surface potential of the form V (x) = gδ(x1)tan π(α · x2 + ω), where for x ∈ Zd we write x = (x1,x2), x1 ∈ Zd1, x2 ∈ Zd2 d2, α ∈ R, ω ∈ [0,1). We first consider the case where the components of the vector α are rationally independent, i.e. ..."
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We study the discrete Schrödinger operator H in Zd with the surface potential of the form V (x) = gδ(x1)tan π(α · x2 + ω), where for x ∈ Zd we write x = (x1,x2), x1 ∈ Zd1, x2 ∈ Zd2 d2, α ∈ R, ω ∈ [0,1). We first consider the case where the components of the vector α are rationally independent, i.e. the case of the quasi periodic potential. We prove that the spectrum of H on the interval [−d,d] (coinciding with the spectrum of the discrete Laplacian) is absolutely continuous. Then we show that generalized eigenfunctions corresponding to this interval have the form of volume (bulk) waves, which are oscillating and non decreasing (or slow decreasing) in all variables. They are the sum of the incident plane wave and of an infinite number of reflected or transmitted plane waves scattered by the ”plane ” Zd2. These eigenfunctions are orthogonal, complete and verify a natural analogue of the LippmannSchwinger equation. We also discuss the case of rational vectors α for d1 = d2 = 1, i.e. a periodic surface potential. In this case we show that the spectrum is absolutely continuous and besides volume (Bloch) waves there are also surface waves, whose amplitude decays exponentially as x1  → ∞. The part of the spectrum corresponding to the surface states consists of a finite number of bands. For large q the bands outside of [−d,d] are exponentially small in q, and converge in a natural sense to the pure point spectrum, that was found in [19] in the case of the Diophantine α’s. 1 1
ABSENCE OF CONTINUOUS SPECTRAL TYPES FOR CERTAIN NONSTATIONARY RANDOM MODELS
, 2004
"... Abstract. We consider continuum random Schrödinger operators of the type Hω = −∆+V0 +Vω with a deterministic background potential V0. We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of − ∆ + V0. The models we treat include ra ..."
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Abstract. We consider continuum random Schrödinger operators of the type Hω = −∆+V0 +Vω with a deterministic background potential V0. We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of − ∆ + V0. The models we treat include random surface potentials as well as sparse or slowly decaying random potentials. In particular, we establish absence of absolutely continuous surface spectrum for random potentials supported near a onedimensional surface (“random tube”) in arbitrary dimension. 1.
ABSENCE OF CONTINUOUS SPECTRAL TYPES FOR CERTAIN NONSTATIONARY RANDOM MODELS IN MEMORY OF
"... Abstract. We consider continuum random Schrödinger operators of the type Hω = − ∆ + V0 + Vω with a deterministic background potential V0. We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of − ∆ + V0. The models we treat includ ..."
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Abstract. We consider continuum random Schrödinger operators of the type Hω = − ∆ + V0 + Vω with a deterministic background potential V0. We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of − ∆ + V0. The models we treat include random surface potentials as well as sparse or slowly decaying random potentials. In particular, we establish absence of absolutely continuous surface spectrum for random potentials supported near a onedimensional surface (“random tube”) in arbitrary dimension. 1.