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Approximation of small-amplitude weakly coupled oscillators with discrete nonlinear Schrodinger equations
Small-amplitude weakly coupled oscillators of the Klein-Gordon lattices are
approximated by equations of the discrete nonlinear Schrodinger type. We show
how to justify this approximation by two methods, which have been very popular
in the recent literature. The first method relies on a priori energy estimates
and multi-scale decompositions. The second method is based on a resonant normal
form theorem. We show that although the two methods are different in the
implementation, they produce equivalent results as the end product. We also
discuss applications of the discrete nonlinear Schrodinger equation in the
context of existence and stability of breathers of the Klein--Gordon lattice
Interpolation inequalities and spectral estimates for magnetic operators
We prove magnetic interpolation inequalities and Keller-Lieb-Thir-ring
estimates for the principal eigenvalue of magnetic Schr{\"o}dinger operators.
We establish explicit upper and lower bounds for the best constants and show by
numerical methods that our theoretical estimates are accurate
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