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ALMOST PERIODIC SCHRÖDINGER OPERATORS II. THE INTEGRATED DENSITY OF STATES
 VOL. 50, NO. DUKE MATHEMATICAL JOURNAL (C) MARCH 1983
, 1983
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On the spectrum of Hecke type operators related to some fractal groups
 TRUDY MAT. INST. STEKLOV
, 1999
"... We give the first example of a connected 4regular graph whose Laplace operator’s spectrum is a Cantor set, as well as several other computations of spectra following a common “finite approximation” method. These spectra are simple transforms of the Julia sets associated to some quadratic maps. The ..."
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Cited by 51 (19 self)
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We give the first example of a connected 4regular graph whose Laplace operator’s spectrum is a Cantor set, as well as several other computations of spectra following a common “finite approximation” method. These spectra are simple transforms of the Julia sets associated to some quadratic maps. The graphs involved are Schreier graphs of fractal groups of intermediate growth, and are also “substitutional graphs”. We also formulate our results in terms of Hecke type operators related to some irreducible quasiregular representations of fractal groups and in terms of the Markovian operator associated to noncommutative dynamical systems via which these fractal groups were originally defined in [Gri80]. In the computations we performed, the selfsimilarity of the groups is reflected in the selfsimilarity of some operators; they are approximated by finite counterparts whose spectrum is computed by an ad hoc factorization process.
The Ten Martini Problem
"... Abstract. We prove the conjecture (known as the \Ten Martini Problem " after Kac and Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all nonzero values of the coupling and all irrational frequencies. 1. ..."
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Abstract. We prove the conjecture (known as the \Ten Martini Problem " after Kac and Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all nonzero values of the coupling and all irrational frequencies. 1.
Schrödinger operators in the twentyfirst century. In:
 Mathematical physics 2000, 283–288, Imp.
, 2000
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Zero Measure Spectrum for the Almost Mathieu Operator
 Commun. Math. Phys
, 1993
"... . We study the almost Mathieu operator: (H ff;;` u)(n) = u(n + 1) + u(n \Gamma 1) + cos(2ßffn + `)u(n) , on l 2 (Z) , and show that for all ; ` , and (Lebesgue) a.e. ff , the Lebesgue measure of its spectrum is precisely j4 \Gamma 2jjj . In particular, for jj = 2 the spectrum is a zero measure c ..."
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Cited by 33 (3 self)
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. We study the almost Mathieu operator: (H ff;;` u)(n) = u(n + 1) + u(n \Gamma 1) + cos(2ßffn + `)u(n) , on l 2 (Z) , and show that for all ; ` , and (Lebesgue) a.e. ff , the Lebesgue measure of its spectrum is precisely j4 \Gamma 2jjj . In particular, for jj = 2 the spectrum is a zero measure cantor set. Moreover, for a large set of irrational ff 's (and jj = 2) we show that the Hausdorff dimension of the spectrum is smaller than or equal to 1=2 . * Work partially supported by the GIF. 1. Introduction In this paper, we study the almost Mathieu (also called Harper's) operator on l 2 (Z) . This is the (bounded, self adjoint) operator H ff;;` , defined by: H ff;;` = H 0 + V ff;;` ; (H 0 u)(n) = u(n + 1) + u(n \Gamma 1) ; (V ff;;` u)(n) = cos(2ßffn + `)u(n) ; (1:1) where ff; ; ` 2 R . H ff;;` is a tight binding model for the Hamiltonian of an electron in a one dimensional lattice, subject to a commensurate (if ff is rational) or incommensurate (if ff is irrational) potential....
Almost everything about the almost Mathieu operator. I
 In XIth International Congress of Mathematical Physics
, 1994
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Cantor spectrum for the almost Mathieu operator
 Commun. Math. Phys
"... In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, (Hb,φx) n = xn+1 + xn−1 + bcos (2πnω + φ) xn on l 2 (Z) and its associated eigenvalue equation to deduce that for b = 0, ±2 and ω Diophantine the spectrum of the operator is a Cantor subset of th ..."
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In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, (Hb,φx) n = xn+1 + xn−1 + bcos (2πnω + φ) xn on l 2 (Z) and its associated eigenvalue equation to deduce that for b = 0, ±2 and ω Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the socalled “Ten Martini Problem ” for these values of b and ω. Moreover, we prove that for b  = 0 small enough or large enough all spectral gaps predicted by the Gap Labelling theorem are open. 1 Introduction. Main
C ∗ algebras and numerical linear algebra
 J. Funct. Anal
"... Abstract. Given a self adjoint operator A on a Hilbert space, suppose that that one wishes to compute the spectrum of A numerically. In practice, these problems often arise in such a way that the matrix of A relative to a natural basis is “sparse”. For example, doubly infinite tridiagonal matrices a ..."
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Abstract. Given a self adjoint operator A on a Hilbert space, suppose that that one wishes to compute the spectrum of A numerically. In practice, these problems often arise in such a way that the matrix of A relative to a natural basis is “sparse”. For example, doubly infinite tridiagonal matrices are usually associated with discretized second order differential operators. In these cases it is easy and natural to compute the eigenvalues of large n × n submatrices of the infinite operator matrix, and to hope that if n is large enough then the resulting distribution of eigenvalues will give a good approximation to the spectrum of A. While this hope is often realized in practice it often fails as well, and it can fail in spectacular ways. The sequence of eigenvalue distributions may not converge as n → ∞, or they may converge to something that has little to do with the original operator A. At another level, even the meaning of ‘convergence ’ has not been made precise in general. In this paper we determine the proper general setting in which one can expect convergence, and we describe the asymptotic behavior of the n × n eigenvalue distrubutions in all but the most pathological cases. Under appropriate
ON RESONANCES AND THE FORMATION OF GAPS IN THE SPECTRUM OF QUASIPERIODIC SCHRÖDINGER EQUATIONS
, 2006
"... n ∈ Z, x, ω ∈ [0, 1] with realanalytic potential function V (x). If L(E, ω0)> 0 for all E ∈ (E′, E ′′) and some Diophantine ω0, then the integrated density of states is absolutely continuous for almost every ω close to ω0, see [GolSch2]. In this work we apply the methods and results of [GolSch2] ..."
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Cited by 18 (2 self)
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n ∈ Z, x, ω ∈ [0, 1] with realanalytic potential function V (x). If L(E, ω0)> 0 for all E ∈ (E′, E ′′) and some Diophantine ω0, then the integrated density of states is absolutely continuous for almost every ω close to ω0, see [GolSch2]. In this work we apply the methods and results of [GolSch2] to establish the formation of a dense set of gaps in ⋃ sp H(x, ω) ∩ (E x ′, E ′′). Our approach is based on multiscale arguments, and is therefore both constructive as well as quantitative. We show how resonances between eigenfunctions of one scale lead to ”pregaps ” at a larger scale. Then we show how these pregaps cannot be filled more than a finite (and uniformly bounded) number of times. To accomplish this, we relate a pregap to pairs of complex zeros of the Dirichlet determinants off the unite circle using the techniques of [GolSch2]. Of basic importance to our entire construction are the finitevolume description of Anderson localization as well as the separation of Dirichlet eigenvalues in a finite volume which were obtained in [GolSch2]. Another essential ingredient is the elimination of triple resonances from Chan [Cha], a special case of which is reproduced here.