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Register a new userLogarithmic Sobolev inequalities and spectral concentration for the cubic Schrodinger equation
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The nonlinear Schrodinger equation NLSE(p, beta), -iu_t=-u_{xx}+beta | u|^{p-2} u=0, arises from a Hamiltonian on infinite-dimensional phase space Lp^2(mT). For pleq 6, Bourgain (Comm. Math. Phys. 166 (1994), 1--26) has shown that there exists a Gibbs measure mu^{beta}_N on balls Omega_N= {phi in Lp^2(mT) : | phi |^2_{Lp^2} leq N} in phase space such that the Cauchy problem for NLSE(p,beta) is well posed on the support of mu^{beta}_N, and that mu^{beta}_N is invariant under the flow. This paper shows that mu^{beta}_N satisfies a logarithmic Sobolev inequality for the focussing case beta <0 and 2leq pleq 4 on Omega_N for all N>0; also mu^{beta} satisfies a restricted LSI for 4leq pleq 6 on compact subsets of Omega_N determined by Holder norms. Hence for p=4, the spectral data of the periodic Dirac operator in Lp^2(mT; mC^2) with random potential phi subject to mu^{beta}_N are concentrated near to their mean values. The paper concludes with a similar result for the spectral data of Hills equation when the potential is random and subject to the Gibbs measure of KdV.
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