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32
Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension
 J. Amer. Math. Soc
"... We consider the nonlinear Schrödinger equation (1.1) i∂tψ + ∂ 2 xψ = −ψ  2σ ψ on the line R with σ> 2. This is exactly the L 2supercritical case and these equations are locally wellposed in H 1 (R) = W 1,2 (R). Let φ = φ(·, α) be the ..."
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Cited by 50 (10 self)
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We consider the nonlinear Schrödinger equation (1.1) i∂tψ + ∂ 2 xψ = −ψ  2σ ψ on the line R with σ> 2. This is exactly the L 2supercritical case and these equations are locally wellposed in H 1 (R) = W 1,2 (R). Let φ = φ(·, α) be the
DISPERSIVE ESTIMATES FOR SCHRÖDINGER OPERATORS IN THE PRESENCE OF A RESONANCE AND/OR AN EIGENVALUE AT ZERO ENERGY IN DIMENSION THREE: I
, 2004
"... ..."
Stable Directions For Excited States Of Nonlinear Schrödinger Equations
 Comm. Partial Differential Equations
, 2001
"... We consider nonlinear Schrödinger equations in R³. Assume that the linear Hamiltonians have two bound states. For certain finite codimension subset in the space of initial data, we construct solutions converging to the excited states in both nonresonant and resonant cases. In the resonant ..."
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Cited by 44 (8 self)
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We consider nonlinear Schrödinger equations in R&sup3;. Assume that the linear Hamiltonians have two bound states. For certain finite codimension subset in the space of initial data, we construct solutions converging to the excited states in both nonresonant and resonant cases. In the resonant case, the linearized operators around the excited states are nonself adjoint perturbations to some linear Hamiltonians with embedded eigenvalues. Although selfadjoint perturbation turns embedded eigenvalues into resonances, this class of nonself adjoint perturbations turn an embedded eigenvalue into two eigenvalues with the distance to the continuous spectrum given to the leading order by the Fermi golden rule.
Dispersive Analysis of Charge Transfer Models
, 2008
"... We prove L p estimates for charge transfer Hamiltonians, including matrix and inhomogeneous generalizations; such equations appear naturally in the study of multisoliton systems. ..."
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Cited by 36 (9 self)
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We prove L p estimates for charge transfer Hamiltonians, including matrix and inhomogeneous generalizations; such equations appear naturally in the study of multisoliton systems.
Nongeneric blowup solutions for the critical focusing
 NLS in 1d, preprint
, 2005
"... We consider the critical focusing NLS in 1d of the form (1.1) i∂tψ + ∂ 2 xψ = −ψ  4 ψ, i = √ −1, ψ = ψ(t, x), and ψ complex valued. It is wellknown that this equation permits standing wave solutions of the form φ(t, x) = e iαt φ0(x, α), α> 0 ..."
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Cited by 28 (9 self)
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We consider the critical focusing NLS in 1d of the form (1.1) i∂tψ + ∂ 2 xψ = −ψ  4 ψ, i = √ −1, ψ = ψ(t, x), and ψ complex valued. It is wellknown that this equation permits standing wave solutions of the form φ(t, x) = e iαt φ0(x, α), α> 0
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 17 (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.
Quantum Brownian Motion in a Simple Model System
, 810
"... Abstract: We consider a quantum particle coupled (with strength λ) to a spatial array of independent noninteracting reservoirs in thermal states (heat baths). Under the assumption that the reservoir correlations decay exponentially in time, we prove that the motion of the particle is diffusive at l ..."
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Cited by 13 (6 self)
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Abstract: We consider a quantum particle coupled (with strength λ) to a spatial array of independent noninteracting reservoirs in thermal states (heat baths). Under the assumption that the reservoir correlations decay exponentially in time, we prove that the motion of the particle is diffusive at large times for small, but finite λ. Our proof relies on an expansion around the kinetic scaling limit (λ ց 0, while time and space scale as λ−2) in which the particle satisfies a Boltzmann equation. We also show an equipartition theorem: the distribution of the kinetic energy of the particle tends to a MaxwellBoltzmann distribution, up to a correction of O(λ2). KEY WORDS: diffusion, kinetic limit, quantum brownian motion
THE CALDERÓN PROBLEM IN TRANSVERSALLY ANISOTROPIC GEOMETRIES
, 2013
"... We consider the anisotropic Calderón problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work [8], it was shown that a metric in a fixed conformal class is uniquely determined by boundary measuremen ..."
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Cited by 11 (4 self)
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We consider the anisotropic Calderón problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work [8], it was shown that a metric in a fixed conformal class is uniquely determined by boundary measurements under two conditions: (1) the metric is conformally transversally anisotropic (CTA), and (2) the transversal manifold is simple. In this paper we will consider geometries satisfying (1) but not (2). The first main result states that the boundary measurements uniquely determine a mixed Fourier transform / attenuated geodesic ray transform (or integral against a more general semiclassical limit measure) of an unknown coefficient. In particular, one obtains uniqueness results whenever the geodesic ray transform on the transversal manifold is injective. The second result shows that the boundary measurements in an infinite cylinder uniquely determine the transversal metric. The first result is proved by using complex geometrical optics solutions involving Gaussian beam quasimodes, and the second result follows from a connection between the Calderón problem and Gel’fand’s inverse problem for the wave equation.