## Operators with singular continuous spectrum, IV: Hausdorff dimensions, rank one pertubations, and localization (1996)

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Venue: | J. Anal. Math |

Citations: | 141 - 31 self |

### BibTeX

@ARTICLE{Rio96operatorswith,

author = {R. Del Rio and N. Makarov and B. Simon},

title = {Operators with singular continuous spectrum, IV: Hausdorff dimensions, rank one pertubations, and localization},

journal = {J. Anal. Math},

year = {1996},

pages = {170}

}

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### Abstract

Abstract. For an operator, A, with cyclic vector ϕ, we study A + λP where P is the rank one projection onto multiples of ϕ. If [α, β] ⊂ spec(A) andA has no a.c. spectrum, we prove that A + λP has purely singular continuous spectrum on (α, β) for a dense Gδ of λ’s. The subject of rank one perturbations of self-adjoint operators and the closely related issue of the boundary condition dependence of Sturm-Liouville operators on [0, ∞) has a long history. We’re interested here in the connection with Borel-Stieltjes transforms of measures (Im z>0):

### Citations

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Perturbation theory for linear operators
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Molchanov: Localization at large disorder and at extreme energies: an elementary derivation
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Citation Context ...‖g‖p < ∞ for some p > 1. Moreover, we could replace the compact support assumption by the finiteness of some moment ∫ |λ| αg(λ) dλ for some α>0. Combining this result with those of Aizenman-Molchanov =-=[2]-=-, we see that the strongly coupled multi-dimensional Anderson model has SULE. §8. Semi-Stability of Dynamical Localization Anderson localization (at least as proven in [1]) implies that if ⃗x is the o... |

118 |
Methods of modern mathematical physics III, Scattering theory
- Reed, Simon
- 1979
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Citation Context ...s a Borel set. This follows, for example, by picking {ϕn} ∞ n=1 an orthonormal basis for L2 (R,dµ), letting F (n ≥ N) be the projection onto the span of {ϕn} ∞ n=N and noting that by the RAGE theorem =-=[25]-=-: { R\S = λ ∣ ∀m lim N →∞ lim K→∞ 1 K ∫K 0 ‖F (n ≥ N)e isAλ ϕm‖ 2 } ds =0 . §6. Rank One Perturbations: Some Examples Rank one perturbations can be described by a measure µ given by (ϕ, (A − z) −1 ∫ d... |

115 | Spectral analysis of rank one perturbations and applications
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- 1993
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Citation Context ...ein), is the idea of using Hausdorff measures and dimensions to classify measures. Our main goal in this paper is to look at the singular spectrum produced by rank one perturbations (and discussed in =-=[7,11,33]-=-) from this point of view. A Borel measure µ is said to have exact dimension α ∈ [0, 1] if and only if µ(S) = 0ifS has dimension β < α and if µ is supported by a set of dimension α. If 0 < α<1, such a... |

91 | T.: Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians
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Citation Context ...x) x − z (1.1) where ρ is a measure with ∫ (|x| +1) −1 dρ(x) < ∞. (1.2) In two fundamental papers Aronszajn [1] and Donoghue [5] related F to spectral theory with important later input by Simon-Wolff =-=[13]-=-. In all three works, as in ours, the function (y real) ∫ dρ(x) G(y) = (x − y) 2 plays an important role. Note we define G to be +∞ if the integral diverges. Note too if G(y) < ∞, then the integral de... |

79 | Quantum Dynamics and decomposition of singular continuous spectra
- Last
- 1996
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Citation Context ...ined by what it isn’t — neither pure point nor absolutely continuous. An important point of view, going back in part to Rodgers and Taylor [27,28], and studied recently within spectral theory by Last =-=[22]-=- (also see references therein), is the idea of using Hausdorff measures and dimensions to classify measures. Our main goal in this paper is to look at the singular spectrum produced by rank one pertur... |

77 |
Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2
- Herman
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Citation Context ...d its translates lie in Uu so U1 is a dense Gδ . Thus, U= Uy n U is a dense G . Π δ Example 1. Consider the Jacobi matrix with V θ {n) = λcos(πβn + Θ) . (1) If λ>2, the Lyapunov exponent is positive (=-=[1, 7]-=-) so if β is irrational, there is no a.c. spectrum for Lebesgue a.e. θ (see e.g. [1]), so h θ has purely singular continuous spectrum for a dense G δ of θ. Sinai [11] and Frohlich-Spencer-Wittwer [4] ... |

75 | Localization at weak disorder: some elementary bounds
- Aizenman
- 1994
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Citation Context ...n goal in this paper is to look at the singular spectrum produced by rank one perturbations (and discussed in [7,11,33]) from this point of view. A Borel measure µ is said to have exact dimension α ∈ =-=[0, 1]-=- if and only if µ(S) = 0ifS has dimension β < α and if µ is supported by a set of dimension α. If 0 < α<1, such a measure is, of necessity, singular continuous. But, there are also singular continuous... |

73 | B.: Almost periodic Schrödinger operators. II: The integrated density of states
- Avron, Simon
- 1983
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Citation Context ... be improved. Our v(n) will have the form v(n) =3cos(παn + θ)+λδn0 (B.2) with α irrational. We’ll prove that α can be constructed so that (B.1) holds for all θ and λ ∈ [0, 1]. It is well known (e.g., =-=[4]-=-) that the Lyapunov exponent, which characterizes solutions of (h − E)u =0fora.e.E,θ, is everywhere larger than or equal to ln( 3 2 ). Thus, by the Simon-Wolff criterion [33,40], (1) and (2) hold for ... |

58 | Operators with singular continuous spectrum: I. General operators
- Simon
(Show Context)
Citation Context ...dense sets. Its complement is thus a dense set by the Baire category theorem. But by Thm. 1, this is precisely {λ|Aλ has no eigenvalues on spec(A0)}, which we conclude is dense. By general principles =-=[12]-=-, it is also a Gδ. Here are some simple corollaries of Thm. 3. We state them in the rank one case but they hold in the cot(θ − θ0) B.C. case also. Corollary 4.1. Suppose that A0 is an operator with no... |

54 |
Souillard: Sur le spectre des opérateurs aux différences finies aléatoires
- Kunz, B
- 1980
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Citation Context ...ng are equivalent: (i) H has SUDL. (ii) H has SULP. (iii) H has SULE. Remarks. 1. The fact that dynamical localization implies point spectrum has a long history, going back at least to Kunz-Souillard =-=[20]-=-. Martinelli-Scoppola [23] used a variant of SULE, which they proved by analysis of eigenfunctions, to prove a restricted form of dynamical localization in the multi-dimensional Anderson model. 2. (ii... |

48 | Singular continuous spectrum for palindromic Schrödinger operators
- Hof, Knill, et al.
- 1995
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Citation Context ...E PERTURBATIONS, AND LOCALIZATION R. del Rio 1 , S. Jitomirskaya 2 ,Y.Last 3 , and B. Simon 3 §1. Introduction Although concrete operators with singular continuous spectrum have proliferated recently =-=[7,11,13,17,34,35,37,39]-=-, we still don’t really understand much about singular continuous spectrum. In part, this is because it is normally defined by what it isn’t — neither pure point nor absolutely continuous. An importan... |

47 |
Theory of the integral
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- 1937
(Show Context)
Citation Context ..., then it converges for small δ. Consider the set { Wα = x ∣ lim δ↓0 µ(x − δ, x + δ) δ α ̸= lim δ↓0 µ(x − δ, x + δ) δα } . (2.3) For α =0,Wα is empty; and for α =1,thetheoremofdelaVallée-Poussin (see =-=[30]-=- or Theorem 7.15 of [29]) says that µ(W1) =0. For0<α<1, however, the situation is quite different: A result going back to Besicovitch [5] (also see Theorem 5.2 of [10]) is that if µ is the restriction... |

46 | Bounded eigenfunctions and absolutely continuous spectra for one dimensional Schrödinger operators
- Simon
- 1996
(Show Context)
Citation Context ...tial (B.2) with λ ∈ [0, 1]. Then we see that ‖˜ Φθ,λ n (E)‖ must also be uniformly bounded. That is, ‖˜ Φθ,λ n (E)‖ <Cfor all n ≥ 0, λ ∈ [0, 1], θ ∈ [0, 2π], and E ∈ Iθ. By, for example, Theorem 2 of =-=[38]-=-, this implies that the imaginary part of the m-function for h(α, θ, λ), which is identical to the Borel transform Fθ,λ of the spectral measure of δ0 (for h(α, θ, λ)), is uniformly bounded. Namely,32... |

38 |
Spectral properties of quantum diffusion on discrete lattices, Europhys
- Guarneri
- 1989
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Citation Context ...ND LOCALIZATION 25 bound on 〈x 2n 〉(t), regardless of whether H has SULE, orevenwhetherH has only pure pointspectrumornot. 2. By a result of Last [22], which extends an idea originaly due to Guarneri =-=[12]-=-, it follows that if the spectral measure of δ0 (for Hλ) is not supported on a set of Hausdorff dimension zero, then for some β>0, lim t −2nβ 〈x 2n 〉(t) > 0. Thus, we get an alternative proof to the f... |

32 |
On the perturbation of spectra
- Donoghue
- 1965
(Show Context)
Citation Context ...nnection with BorelStieltjes transforms of measures (Im z>0): F (z) = ∫ dρ(x) x − z (1.1) where ρ is a measure with ∫ (|x| +1) −1 dρ(x) < ∞. (1.2) In two fundamental papers Aronszajn [1] and Donoghue =-=[5]-=- related F to spectral theory with important later input by Simon-Wolff [13]. In all three works, as in ours, the function (y real) ∫ dρ(x) G(y) = (x − y) 2 plays an important role. Note we define G t... |

29 |
Lyapunov exponents and spectra for one-dimensional random Schrödinger operators
- Kotani
- 1984
(Show Context)
Citation Context ...us (a) follows from Thm. 2. To prove (b), note that if E0 has a solution obeying (5.1–2) and E0 is not an ∞∫ eigenvalue of Hθ0, then lim |G(0,x; E + iɛ)| ɛ↓0 0 2 dx < ∞. Now apply the ideas of Kotani =-=[11]-=- and Simon-Wolff [13]. (c) follows from Thm. 3. Example 5.2. Suppose that [a,b] ⊂ spec(− d2 dx2 + V (x)) and that for a.e. E ⊂ 1 [a,b], lim x→∞ |x| ln‖TE(x)‖ = γ(E) andispositive. HereTisthe standard ... |

26 |
Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials
- Sorets, Spencer
- 1991
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Citation Context ...odinger case with V θ (x)= — fc[cos(2πx) + cos(2π/?x + 0)]. Then, Frδhlich-Spencer — Wittwer [4] have proven for a.e. θ (k large enough), there is pure point spectrum for low energies. Sorets-Spencer =-=[12]-=- have proven positivity of the Lyapunov exponent for a wider area of low energy. We conclude that for a dense G δ of 0, there is purely singular continuous spectrum for low energies. 2. Proof of Theor... |

25 |
On a problem of Weyl in the theory of singular Sturm-Liouville equations
- Aronszajn
- 1957
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Citation Context ...ed here in the connection with BorelStieltjes transforms of measures (Im z>0): F (z) = ∫ dρ(x) x − z (1.1) where ρ is a measure with ∫ (|x| +1) −1 dρ(x) < ∞. (1.2) In two fundamental papers Aronszajn =-=[1]-=- and Donoghue [5] related F to spectral theory with important later input by Simon-Wolff [13]. In all three works, as in ours, the function (y real) ∫ dρ(x) G(y) = (x − y) 2 plays an important role. N... |

24 |
Fourier asymptotics of fractal measures
- Strichartz
- 1990
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Citation Context ...inally, pick αm+1 so |αn − αm+1| < ∆n for n =1,...,m. Remarks. 1. (B.10) is the standard estimate for which ||| · ||| was introduced (see (X.4.18) of [18]). It is used here as the Strichartz estimate =-=[41]-=- is used in the proof of Theorem 6.1 of [22]. Indeed, the above proof is essentially a variant of the proof of a similar result in [22] (Theorem 7.2 of [22]). 2. One can similarly prove an analogous r... |

23 |
A pure point spectrum of the stochastic one-dimensional schrödinger operator
- Goldsheid, Pastur
- 1977
(Show Context)
Citation Context ...E in an interval. Results of this genre have been found previously by Goldsheid [6] and Carmona [2]. Example 5.3. Consider a one-dimensional random model with localization, for example, the GMP model =-=[7,2]-=-. Then for almost every E in [α, ∞), one knows γ(E) exists and is positive. It follows from Thm. 5.1 that for a locally uncountable set of boundary conditions (a Lebesgue typical set), one has pure po... |

22 | Zero measure spectrum for the almost Mathieu operator
- Last
- 1994
(Show Context)
Citation Context ... ^15. In that case there are intertwined locally uncountable sets of θ with only pure point and with only singular continuous spectrum. For λ = 2, spec(/z θ ) has zero measure for many irrational /Γs =-=[9]-=- and so no a.c. spectrum. We conclude Theorem 3. For the example (1), h θ has purely singular continuous spectrum for a dense G δ ofθ's if β is irrational and λ>2 or if the continued fraction expansio... |

22 |
Bounded solutions and absolute continuity of Sturm-Liouville operators
- Stolz
- 1992
(Show Context)
Citation Context ...ns (a Baire typical set), one has singular spectrum. Each spectral type is unstable to change to the other spectral type. Example 5.4. Let H = − d2 dx2 +cos( √ x)onL2 (0, ∞), a model studied by Stolz =-=[14]-=-. As proven by him for any boundary condition θ: spec(Hθ) =[−1, ∞). spec(Hθ) is purely absolutely continuous on (1, ∞). Kirsch et al. [10] prove that for a.e. θ, Hθ has pure point spectrum in [−1, 1] ... |

21 |
Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential
- Sinaĭ
- 1987
(Show Context)
Citation Context ... Lyapunov exponent is positive ([1, 7]) so if β is irrational, there is no a.c. spectrum for Lebesgue a.e. θ (see e.g. [1]), so h θ has purely singular continuous spectrum for a dense G δ of θ. Sinai =-=[11]-=- and Frohlich-Spencer-Wittwer [4] have proven for λ large and β having good Diophantine properties, a.e. θ has pure point spectrum, and Jitomirskaya [8] has proven that for λ ^15. In that case there a... |

21 | E.: Introduction to the mathematical theory of Anderson localization - Martinelli, Scoppola - 1987 |

21 | Absence of ballistic motion
- Simon
- 1990
(Show Context)
Citation Context ...nsideration in this paper are a significant desideratum. One can modify the proof to replace F (t) byt2 /f(t) for any monotone f with lim f(t) =∞. Thus, this example also shows that the result of t→∞ =-=[31]-=- that point spectrum implies lim t→∞ ‖xe−ithδ0‖ 2/ t 2 =0 cannot be improved. Our v(n) will have the form v(n) =3cos(παn + θ)+λδn0 (B.2) with α irrational. We’ll prove that α can be constructed so tha... |

20 |
A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants
- Last
- 1993
(Show Context)
Citation Context ... from 0 to n for the potential (B.2). That is, Φθ n (E) ≡ T θ n (E)T θ n−1 (E) ... Tθ 0 (E), where T θ n (E) ≡ ( ) E − v(n) −1 1 0 and v(n) is given by (B.2) with λ = 0. By, for example, Lemma 3.1 of =-=[21]-=-, we have the bound ‖Φθ mq−1(E)‖ ≤2q ∣ ∂Eθ ∣ ∂k −1 for any integer m>0. (Remark: Lemma 3.1 of [21] is formulated for the transfer matrix over one period, but it is easy to see from its proof that the ... |

18 |
Localization for a class of one-dimensional quasi-periodic Schrödinger operators
- Fröhlich, Spencer, et al.
- 1990
(Show Context)
Citation Context ..., 7]) so if β is irrational, there is no a.c. spectrum for Lebesgue a.e. θ (see e.g. [1]), so h θ has purely singular continuous spectrum for a dense G δ of θ. Sinai [11] and Frohlich-Spencer-Wittwer =-=[4]-=- have proven for λ large and β having good Diophantine properties, a.e. θ has pure point spectrum, and Jitomirskaya [8] has proven that for λ ^15. In that case there are intertwined locally uncountabl... |

18 |
Pure point spectrum under 1-parameter perturbations and instability of Anderson localization
- Gordon
- 1994
(Show Context)
Citation Context ...E PERTURBATIONS, AND LOCALIZATION R. del Rio 1 , S. Jitomirskaya 2 ,Y.Last 3 , and B. Simon 3 §1. Introduction Although concrete operators with singular continuous spectrum have proliferated recently =-=[7,11,13,17,34,35,37,39]-=-, we still don’t really understand much about singular continuous spectrum. In part, this is because it is normally defined by what it isn’t — neither pure point nor absolutely continuous. An importan... |

16 | Anderson localization for the almost mathieu equation II, point spectrum for A > 2
- Jitomirskaya
- 1995
(Show Context)
Citation Context ...ontinuous spectrum for a dense G δ of θ. Sinai [11] and Frohlich-Spencer-Wittwer [4] have proven for λ large and β having good Diophantine properties, a.e. θ has pure point spectrum, and Jitomirskaya =-=[8]-=- has proven that for λ ^15. In that case there are intertwined locally uncountable sets of θ with only pure point and with only singular continuous spectrum. For λ = 2, spec(/z θ ) has zero measure fo... |

12 | Singular continuous spectrum is generic
- Rio, Jitomirskaya, et al.
(Show Context)
Citation Context ... good Diophantine properties. 1. Introduction This is a paper that provides yet another place where singular continuous spectrum occurs in the theory of Schrodinger operators and Jacobi matrices (see =-=[5,6,2,10,3]-=-). It is especially interesting because it will provide examples where a non-resonance condition in a KAM argument is not merely needed for technical reasons but necessary. Our main results, proven in... |

12 |
p norms of the Borel transform and the decomposition of measures
- Simon, L
- 1995
(Show Context)
Citation Context ...ND B. SIMON Theorem 3.5. Suppose that sup ɛ>0 ɛ s ∫b a |Im Fµ(x + iɛ)| 2 dx < ∞ for some s<1. Thenµ (a, b) gives zero weight to sets of dimension less than 1 − s. Remark. The s = 0 result is stronger =-=[36]-=-; in that case µ is purely absolutely continuous on (a, b). Proof. We’ll prove that for any β<1 − s and any closed interval I ⊂ (a, b), we have ∫ dµ(x) dµ(y) < ∞. (3.3) |x − y| β x∈I y∈I This implies ... |

11 |
A certain inverse problem for Sturm-Liouville operators
- Javrjan
- 1971
(Show Context)
Citation Context ...) ≤ (E(|f|)) 1/(2−s) Now take E’s. Since ∫ and E(dµλ(E)) ≤‖g‖∞ dλ(dµλ(E)) = ‖g‖∞ dE where the last equality is a result explicitly in Simon-Wolff [40] but obtained in related forms earlier by Javrjan =-=[15]-=- and Kotani [19]. Proof of Aizenman’s Theorem (Theorem 7.7). The hypothesis (7.10) implies that for a.e. pairs ω,λ ∈ [a, b] so for a.e. such pairs, |(δn, (Hω − λ − i0) −1 δm)| ≤Cω,λ,me −µ|n−m|/2 ‖(Hω ... |

10 |
On the comparison of transcendents, with certain applications to the theory of definite integrals
- Boole
(Show Context)
Citation Context ...−1sHAUSDORFF DIMENSIONS, RANK ONE PERTURBATIONS, AND LOCALIZATION 29 so (A.9) implies |St| = t−1 . Remarks. 1. Boole’s equality for µ, a measure with a finite number of pure points, was found in 1857 =-=[6]-=-. See [1,24] for more recent history. 2. Using this result in this form, it is not hard to show for any measure µ, lim t|{x ||F(x + i0)| >t}| =2µsing(R) t→∞ the mass of the singular part of µ. Boole’s... |

10 |
Additive set functions in Euclidean space
- Rogers, Taylor
- 1963
(Show Context)
Citation Context ...tinuous spectrum. In part, this is because it is normally defined by what it isn’t — neither pure point nor absolutely continuous. An important point of view, going back in part to Rodgers and Taylor =-=[27,28]-=-, and studied recently within spectral theory by Last [22] (also see references therein), is the idea of using Hausdorff measures and dimensions to classify measures. Our main goal in this paper is to... |

7 |
Exponential localization in one-dimensional disordered systems
- Carmona
- 1982
(Show Context)
Citation Context ...here either lim 1 |x| ‖TE(x)‖ fails to exist or is zero. Thus, a positive limit can never exist for all E in an interval. Results of this genre have been found previously by Goldsheid [6] and Carmona =-=[2]-=-. Example 5.3. Consider a one-dimensional random model with localization, for example, the GMP model [7,2]. Then for almost every E in [α, ∞), one knows γ(E) exists and is positive. It follows from Th... |

7 |
Cyclic vectors in the Anderson model
- Simon
(Show Context)
Citation Context ... Theorem 7.5 hold in the context of the Anderson model. We’re dealing here with models depending on a random parameter so we first reduce SUDL to a requirement on expectations. General considerations =-=[32]-=- imply that the spectrum is simple in the localized regime.HAUSDORFF DIMENSIONS, RANK ONE PERTURBATIONS, AND LOCALIZATION 23 Theorem 7.6. Let (Ω,µ) be a probability measure space and E( · ) its expec... |

6 |
B.: Almost periodic Schrόdinger operators, II. The integrated density of states
- Avron, Simon
- 1983
(Show Context)
Citation Context ...d its translates lie in Uu so U1 is a dense Gδ . Thus, U= Uy n U is a dense G . Π δ Example 1. Consider the Jacobi matrix with V θ {n) = λcos(πβn + Θ) . (1) If λ>2, the Lyapunov exponent is positive (=-=[1, 7]-=-) so if β is irrational, there is no a.c. spectrum for Lebesgue a.e. θ (see e.g. [1]), so h θ has purely singular continuous spectrum for a dense G δ of θ. Sinai [11] and Frohlich-Spencer-Wittwer [4] ... |

5 |
On a problem of Weyl in the theory of Sturm-Liouville equations
- Aronszajn
- 1957
(Show Context)
Citation Context ...(|x| +1) −1 dµ(x) < ∞, we define its Borel transform by Fµ(z) = ∫ dµ(x) x − z for Im z>0. These play a crucial role in the theory of rank one perturbations as originally noticed by Aronszajn-Donoghue =-=[3,9]-=-; see [33] for their properties and this theory. In this section, we’ll translate Theorem 2.1 into Borel transform language. Definition. Fix γ ≤ 1andx. Let Our goal in this section is to prove: Q γ µ(... |

4 |
On linear sets of points of fractional dimensions
- Besicovitch
- 1929
(Show Context)
Citation Context ...Wα is empty; and for α =1,thetheoremofdelaVallée-Poussin (see [30] or Theorem 7.15 of [29]) says that µ(W1) =0. For0<α<1, however, the situation is quite different: A result going back to Besicovitch =-=[5]-=- (also see Theorem 5.2 of [10]) is that if µ is the restriction of h α to a set of finite positive h α -measure, then µ is supported on Wα. Moreover, there are even examples of µ’s where for a.e. x w.... |

4 |
On the distributions of boundary values of Cauchy integrals
- Poltoratski
- 1996
(Show Context)
Citation Context ...RFF DIMENSIONS, RANK ONE PERTURBATIONS, AND LOCALIZATION 29 so (A.9) implies |St| = t−1 . Remarks. 1. Boole’s equality for µ, a measure with a finite number of pure points, was found in 1857 [6]. See =-=[1,24]-=- for more recent history. 2. Using this result in this form, it is not hard to show for any measure µ, lim t|{x ||F(x + i0)| >t}| =2µsing(R) t→∞ the mass of the singular part of µ. Boole’s equality ap... |

3 |
Exceptional values of the boundary phase for the Schrödinger equation on the semi-axis
- Gordon
- 1992
(Show Context)
Citation Context ...eigenvalues in spec(A0)} (resp. {θ|Hθ has no eigenvalues in spec(Hθ0} is a dense Gδ in R (resp. [0, 2π])). While not stated precisely in those terms, Thm. 2 is a generalization of del Rio [4]. Gordon =-=[8,9]-=- has independently obtained these results by different methods. Thm. 2 is quite easy and appears in §2. Thm. 3 is deeper and depends on some subtle estimates of F found in §3 and applied in §4 toprove... |

3 |
One dimensional wave equations in disordered media
- Delyon, Kunz, et al.
- 1983
(Show Context)
Citation Context ...l e -~lm-ql . t180 R. DEL RIO, S. JITOMIRSKAYA, Y. LAST AND B. SIMON The result now follows from the trivial bound (1 + x) ~ <_ u~e'Xe -~ for e < 1. [] So when does (7.9) hold? Delyon-Kunz-Souillard =-=[8]-=- have proven this bound for a general class of one-dimensional random potentials. In general, we have the following beautiful bound of Aizenman: Theorem 7.7 (Aizenman's theorem) Let V~(n) be a family ... |

3 |
Singular continuous spectrum and uniform localization for ergodic SchrOdinger operators
- Jitomirskaya
(Show Context)
Citation Context ...atH on ℓ 2 (Z ν )hasULE if there are C, α > 0with |ϕn(m)| ≤Ce −α|m−mn| (C.1)34 R. DEL RIO, S. JITOMIRSKAYA, Y. LAST, AND B. SIMON for all eigenfunctions ϕn and suitable mn. Motivated by Jitomirskaya =-=[16]-=-, we present a simple argument that many models do not have ULE: Let Ω be a topological space, Ti :Ω→ Ω, i =1,...,ν commuting homeomorphisms, and let µ be an ergodic Borel measure on Ω. Let f :Ω→ R be... |

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Hausdorff measures
- Rodgers
- 1970
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Citation Context ...e’ll sometimes write αµ(x0) if we want to be explicit about the µ involved; and if we have a one-parameter family µλ, we’lluseαλ for αµλ. The following is a result of Rodgers-Taylor [27,28] (also see =-=[26]-=-):4 R. DEL RIO, S. JITOMIRSKAYA, Y. LAST, AND B. SIMON Theorem 2.1. Let µ be any measure and α ∈ [0, 1]. LetTα = {x | Dα µ(x) =∞} and let χα be its characteristic function. Let dµαs = χα dµ and dµαc ... |

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The analysis of additive set functions in Euclidean space
- Rodgers, Taylor
- 1959
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Citation Context ...tinuous spectrum. In part, this is because it is normally defined by what it isn’t — neither pure point nor absolutely continuous. An important point of view, going back in part to Rodgers and Taylor =-=[27,28]-=-, and studied recently within spectral theory by Last [22] (also see references therein), is the idea of using Hausdorff measures and dimensions to classify measures. Our main goal in this paper is to... |

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Asymptotics of the product of random matrices depending on a parameter
- Goldsheid
- 1975
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Citation Context ... dense Gδ of E where either lim 1 |x| ‖TE(x)‖ fails to exist or is zero. Thus, a positive limit can never exist for all E in an interval. Results of this genre have been found previously by Goldsheid =-=[6]-=- and Carmona [2]. Example 5.3. Consider a one-dimensional random model with localization, for example, the GMP model [7,2]. Then for almost every E in [α, ∞), one knows γ(E) exists and is positive. It... |

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point spectrum under 1-parameter perturbations and instability of Anderson localization
- Pure
(Show Context)
Citation Context ...eigenvalues in spec(A0)} (resp. {θ|Hθ has no eigenvalues in spec(Hθ0} is a dense Gδ in R (resp. [0, 2π])). While not stated precisely in those terms, Thm. 2 is a generalization of del Rio [4]. Gordon =-=[8,9]-=- has independently obtained these results by different methods. Thm. 2 is quite easy and appears in §2. Thm. 3 is deeper and depends on some subtle estimates of F found in §3 and applied in §4 toprove... |