Green's Function Estimates for Lattice Schrödinger Operators and Applications.

By Jean Bourgain. 2004

482 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a63c62e87a95ee35726b34a4c3efd277b9311869

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Citations in the corpus

An Abstract Nash–Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds (2013, methodology): “In the case of NLS and NLW on Tn these estimates have been obtained in [4, 5] via a multiscale analysis on Sobolev spaces (see [10] in an analytic setting).”
Quasi-periodic solutions in a nonlinear Schrödinger equation (2007, methodology): “Using the CWB method, similar existence results of quasiperiodic solutions were shown by Bourgain [6,8] with respect to the periodic boundary condition.”
KAM for autonomous quasi-linear perturbations of mKdV (2014, unknown): “For PDEs in higher space dimension the theory has been more recently extended by Bourgain [10], Eliasson–Kuksin [13], and Berti–Bolle [6], Geng–Xu–You [14], Procesi–Procesi [30,29], Wang [32].”
A KAM theorem for one dimensional Schrödinger equation with periodic boundary conditions (2005, methodology): “Remarkably, in[6], Bourgain constructed the quasi-periodic solutions of nonlinear Schrödinger equation and nonlinear wave equation in arbitrary space dimension by controlling Green’s functions relying on more powerful methods than those in[5].”
KAM for Reversible Derivative Wave Equations (2012, methodology): “It was however remarked by Bourgain that the construction of periodic and quasi-periodic solutions using the Lyapunov-Schmidt decomposition and the Newton iteration method of Craig-Wayne [14] and [8][11] is a-priori not restricted to Hamiltonian systems.”
Branching of Cantor Manifolds of Elliptic Tori and Applications to PDEs (2011, background): “Craig-Wayne [14], Bourgain [8]-[10], Berti-Bolle [5] and with Procesi [6].”
Quasi-periodic solutions with Sobolev regularity of NLS on $\mathbb {T}^d$ with a multiplicative potential (2013, methodology): “The improvement in [13] comes from the use of sophisticated techniques developed in the context of Anderson localization theory in Bourgain-Goldstein-Schlag [14], Bourgain [11], see also Bourgain-Wang [15].”
Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation (2010, methodology): “Bourgain used refinements [19] of the Strichartz inequality to prove polynomial-in-time upper bounds [16] on Sobolev norm growth.”
A KAM Theorem for Hamiltonian Partial Differential Equations in Higher Dimensional Spaces (2006, result): “And although the existence results are not new, since Bourgain has obtained the results for more general classes of equations [ 6 ], the KAM approach may provide more information about the constructed solutions.”
EQUILIBRIUM MEASURES AND CAPACITIES IN SPECTRAL THEORY (2007, background): “For examples with such coexistent spectrum (some only worked for the continuum case), see [17, 18, 36, 37].”
Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces (2011, methodology): “[ 10 ]. Moreover, also the KAM approach has been extended by Eliasson and Kuksin [18] to prove the existence of elliptic tori for Schr odinger equations on Tn with a convolution potential.”
Growth of Sobolev Norms of Solutions of Linear Schrödinger Equations on Some Compact Manifolds (2009, background): “Let us state the following lemma due to Bourgain ([4] Lemma 8.1; see also for the proof lemma 19.10 in [6]): We denote for α ∈ A”
Quasi-periodic solutions with Sobolev regularity of NLS on T^d with a multiplicative potential (2010, methodology): “The improvement in [13] comes from the use of sophisticated techniques developed in the context of Anderson localization theory in Bourgain-Goldstein-Schlag [14], Bourgain [11], see also Bourgain-Wang [15].”
Global theory of one-frequency Schrödinger operators (2015, background): “For small analytic potentials, it follows from the work of BourgainJitomirskaya ([BJ1], [BJ2]; see also [AJ]) that the Lyapunov exponent is zero (and hence constant) in the spectrum.”