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Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles
 Arn61] [AS83] V. I. Arnol ′ d. Small denominators. I. Mapping the
, 2003
"... Abstract. We show that for almost every frequency α ∈ R \ Q, for every C ω potential v: R/Z → R, and for almost every energy E the corresponding quasiperiodic Schrödinger cocycle is either reducible or nonuniformly hyperbolic (similar results are valid in the smooth category). We describe several a ..."
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Cited by 26 (5 self)
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Abstract. We show that for almost every frequency α ∈ R \ Q, for every C ω potential v: R/Z → R, and for almost every energy E the corresponding quasiperiodic Schrödinger cocycle is either reducible or nonuniformly hyperbolic (similar results are valid in the smooth category). We describe several applications for the quasiperiodic Schrödinger operator, including persistence of absolutely continuous spectrum under perturbations of the potential. Such results also allow us to complete the proof of the AubryAndré conjecture on the measure of the spectrum of the Almost Mathieu Operator.
Existence and uniqueness of the integrated density of states for Schrödinger operators with magnetic fields and unbounded random potentials, Rev
 Math. Phys
"... The object of the present study is the integrated density of states of a quantum particle in multidimensional Euclidean space which is characterized by a Schrödinger operator with a constant magnetic field and a random potential which may be unbounded from above and from below. For an ergodic rando ..."
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Cited by 23 (7 self)
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The object of the present study is the integrated density of states of a quantum particle in multidimensional Euclidean space which is characterized by a Schrödinger operator with a constant magnetic field and a random potential which may be unbounded from above and from below. For an ergodic random potential satisfying a simple moment condition, we give a detailed proof that the infinitevolume limits of spatial eigenvalue concentrations of finitevolume operators with different boundary conditions exist almost surely. Since all these limits are shown to coincide with the expectation of the trace of the spatially localized spectral family of the infinitevolume operator, the integrated density of states is almost surely nonrandom and independent of the chosen boundary condition. Our proof of the
Lyapunov exponents and spectral analysis of ergodic Schrödinger operators: A survey of Kotani theory and its applications
 in “Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday
, 2007
"... Abstract. The absolutely continuous spectrum of an ergodic family of onedimensional Schrödinger operators is completely determined by the Lyapunov exponent as shown by Ishii, Kotani and Pastur. Moreover, the part of the theory developed by Kotani gives powerful tools for proving the absence of absol ..."
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Cited by 13 (4 self)
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Abstract. The absolutely continuous spectrum of an ergodic family of onedimensional Schrödinger operators is completely determined by the Lyapunov exponent as shown by Ishii, Kotani and Pastur. Moreover, the part of the theory developed by Kotani gives powerful tools for proving the absence of absolutely continuous spectrum, the presence of absolutely continuous spectrum, and even the presence of purely absolutely continuous spectrum. We review these results and their recent applications to a number of problems: the absence of absolutely continuous spectrum for rough potentials, the absence of absolutely continuous spectrum for potentials defined by the doubling map on the circle, and the absence of singular spectrum for the subcritical almost Mathieu operator. 1.
Some bound state problems in quantum mechanics
 Proc. Symp. Pure Math., 76.1, American Mathematical Society , in “Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday
, 2007
"... We give a review of semiclassical estimates for bound states and their eigenvalues for Schrödinger operators. Motivated by the classical results, we discuss their recent improvements for single particle Schrödinger operators as well as some applications of these semiclassical bounds to multipart ..."
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Cited by 11 (0 self)
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We give a review of semiclassical estimates for bound states and their eigenvalues for Schrödinger operators. Motivated by the classical results, we discuss their recent improvements for single particle Schrödinger operators as well as some applications of these semiclassical bounds to multiparticle systems, in particular, large atoms and the stability of matter.
Imbedded Singular Continuous Spectrum For Schrödinger Operators
 Evolutionary Algorithms in Engineering Applications
, 1997
"... We construct examples of potentials V (x) satisfying jV (x)j where the function h(x) is growing arbitrarily slowly, such that the corresponding Schrödinger operator has imbedded singular continuous spectrum. This solves one of the fifteen "twentyfirst century" problems for Schrödinger operators pos ..."
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Cited by 11 (3 self)
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We construct examples of potentials V (x) satisfying jV (x)j where the function h(x) is growing arbitrarily slowly, such that the corresponding Schrödinger operator has imbedded singular continuous spectrum. This solves one of the fifteen "twentyfirst century" problems for Schrödinger operators posed by Barry Simon in [22]. The construction also provides the first example of a Schrödinger operator for which Möller wave operators exist but are not asymptotically complete due to the presence of singular continuous spectrum.
Almost localization and almost reducibility
"... Abstract. We develop a quantitative version of Aubry duality and use it to obtain several sharp estimates for the dynamics of Schrödinger cocycles associated to a nonperturbatively small analytic potential and Diophantine frequency. In particular, we establish the full version of Eliasson’s reducib ..."
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Cited by 6 (3 self)
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Abstract. We develop a quantitative version of Aubry duality and use it to obtain several sharp estimates for the dynamics of Schrödinger cocycles associated to a nonperturbatively small analytic potential and Diophantine frequency. In particular, we establish the full version of Eliasson’s reducibility theory in this regime (our approach actually leads to improvements even in the perturbative regime: we are able to show, for all energies, “almost reducibility ” in some band of analyticity). We also prove 1/2Hölder continuity of the integrated density of states. For the almost Mathieu operator, our results hold through the entire regime of subcritical coupling and imply also the dry version of the Ten Martini Problem for the concerned parameters. 1.
Finite gap potentials and WKB asymptotics for onedimensional Schrödinger operators
 COMM. MATH. PHYS
, 2001
"... Consider the Schrödinger operator H = −d 2 /dx 2 + V (x) with powerdecaying potential V (x) = O(x −α). We prove that a previously obtained dimensional bound on exceptional sets of the WKB method is sharp in its whole range of validity. The construction relies on pointwise bounds on finite gap pot ..."
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Cited by 5 (1 self)
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Consider the Schrödinger operator H = −d 2 /dx 2 + V (x) with powerdecaying potential V (x) = O(x −α). We prove that a previously obtained dimensional bound on exceptional sets of the WKB method is sharp in its whole range of validity. The construction relies on pointwise bounds on finite gap potentials. These bounds are obtained by an analysis of the Jacobi inversion problem on hyperelliptic Riemann surfaces.
Absolute continuity of the integrated density of states for the almost Mathieu operator with noncritical coupling, preprint
"... Abstract. We show that the integrated density of states of the almost Mathieu operator is absolutely continuous if and only if the coupling is noncritical. We deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase, settling the measuretheoretical c ..."
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Cited by 5 (3 self)
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Abstract. We show that the integrated density of states of the almost Mathieu operator is absolutely continuous if and only if the coupling is noncritical. We deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase, settling the measuretheoretical case of Problem 6 of Barry Simon’s list of Schrödinger operator problems for the twentyfirst century. 1.
The Ionization Conjecture in HartreeFock Theory
, 2000
"... We prove the ionization conjecture within the HartreeFock theory of atoms. More precisely, we prove that, if the nuclear charge is allowed to tend to infinity, the maximal negative ionization charge and the ionization energy of atoms nevertheless remain bounded. Moreover, we show that in Hartre ..."
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Cited by 5 (1 self)
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We prove the ionization conjecture within the HartreeFock theory of atoms. More precisely, we prove that, if the nuclear charge is allowed to tend to infinity, the maximal negative ionization charge and the ionization energy of atoms nevertheless remain bounded. Moreover, we show that in HartreeFock theory the radius of an atom (properly defined) is bounded independently of its nuclear charge.
Absolutely continuous spectrum for the almost Mathieu operator
, 2008
"... We prove that the spectrum of the almost Mathieu operator is absolutely continuous if and only if the coupling is subcritical. This settles Problem 6 of Barry Simon’s list of Schrödinger operator problems for the twentyfirst century. ..."
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Cited by 5 (3 self)
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We prove that the spectrum of the almost Mathieu operator is absolutely continuous if and only if the coupling is subcritical. This settles Problem 6 of Barry Simon’s list of Schrödinger operator problems for the twentyfirst century.