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Time decay for solutions of Schrödinger equations with rough and timedependent potentials.
, 2001
"... In this paper we establish dispersive estimates for solutions to the linear SchrSdinger equation in three dimension (0.1) 1.0tO  A0 + Vb = 0, O(s) = f where V(t, x) is a timedependent potential that satisfies the conditions suPllV(t,.)llL(R) + sup f f IV(*'x)l drdy < Co. ..."
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Cited by 118 (14 self)
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In this paper we establish dispersive estimates for solutions to the linear SchrSdinger equation in three dimension (0.1) 1.0tO  A0 + Vb = 0, O(s) = f where V(t, x) is a timedependent potential that satisfies the conditions suPllV(t,.)llL(R) + sup f f IV(*'x)l drdy < Co.
On the Absolutely Continuous Spectrum of OneDimensional Schrödinger Operators with Square Summable Potentials
 Comm. Math. Phys
, 1999
"... Abstract: For continuous and discrete onedimensional Schrödinger operators with square summable potentials, the absolutely continuous part of the spectrum is essentially supported by T0;1 / and T−2; 2U respectively. This fact is proved by considering a priori estimates for the transmission coeffici ..."
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Cited by 95 (6 self)
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Abstract: For continuous and discrete onedimensional Schrödinger operators with square summable potentials, the absolutely continuous part of the spectrum is essentially supported by T0;1 / and T−2; 2U respectively. This fact is proved by considering a priori estimates for the transmission coefficient. 1.
Global stability of vortex solutions of the twodimensional NavierStokes equation
 Comm. Math. Phys
"... NavierStokes equation ..."
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Spectra of selfadjoint extensions and applications to solvable Schrödinger operators
, 2007
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Moment Analysis for Localization in Random Schrödinger Operators
, 2005
"... We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the ..."
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Cited by 62 (15 self)
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We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonancediffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the LifshitzKrein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weakL¹ estimate concerning the boundaryvalue distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.
Solitary wave dynamics in an external potential
 Comm. Math. Phys
"... We study the behavior of solitarywave solutions of some generalized nonlinear Schrödinger equations with an external potential. The equations have the feature that in the absence of the external potential, they have solutions describing inertial motions of stable solitary waves. We construct soluti ..."
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Cited by 60 (7 self)
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We study the behavior of solitarywave solutions of some generalized nonlinear Schrödinger equations with an external potential. The equations have the feature that in the absence of the external potential, they have solutions describing inertial motions of stable solitary waves. We construct solutions of the equations with a nonvanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that, over a large interval of time, they describe a solitary wave whose center of mass motion is a solution of Newton’s equations of motion for a point particle in the given external potential, up to small corrections corresponding to radiation damping. 1
Magnetic Bottles in Connection With Superconductivity
, 2001
"... Motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two, a lot of papers devoted to the analysis in a semiclassical regime of the lowest eigenvalue of the Schrodinger operator with magnetic field have appeared recently. Here ..."
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Cited by 60 (17 self)
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Motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two, a lot of papers devoted to the analysis in a semiclassical regime of the lowest eigenvalue of the Schrodinger operator with magnetic field have appeared recently. Here we would like to mention the works by BernoffSternberg, LuPan and Del PinoFelmerSternberg. This recovers partially questions analyzed in a different context by the authors around the question of the so called magnetic bottles. Our aim is to analyze the former results, to treat them in a more systematic way and to improve them by giving sharper estimates of the remainder. In particular, we improve significatively the lower bounds and as a byproduct we solve a conjecture proposed by BernoffSternberg concerning the localization of the ground state inside the boundary in the case with constant magnetic fields.