Pair excitations and the mean field approximation of interacting Bosons, II

We consider a large number of Bosons with interaction potential $v_N(x)=N^{3 \beta}v(N^{\beta}x)$. In earlier papers we considered a set of equations for the condensate $\phi$ and pair excitation function $k$ and proved that they provide a Fock space approximation to the exact evolution of the condensate for $\beta <\frac{1}{3}$. This result was extended to the case $\beta<\frac{1}{2}$ by E. Kuz, where it was also argued informally that the equations of our earlier work do not provide an approximation for $\beta>\frac{1}{2}$. In 2013, we introduced a coupled refinement of our original equations and conjectured that they provide a Fock space approximation in the range $0 \le \beta < 1$. In the current paper we prove that this is indeed the case for $\beta < \frac{2}{3}$, at least locally in time. In order to do that, we re-formulate the equations of \cite{GMM} in a way reminiscent of BBGKY and apply harmonic analysis techniques in the spirit of X. Chen and J. Holmer to prove the necessary estimates. In turn, these estimates provide bounds for the pair excitation function $k$

Collapsing Estimates and the Rigorous Derivation of the 2d Cubic Nonlinear Schr\"odinger Equation with Anisotropic Switchable Quadratic Traps

We consider the 2d and 3d many body Schr\"odinger equations in the presence of anisotropic switchable quadratic traps. We extend and improve the collapsing estimates in Klainerman-Machedon [24] and Kirkpatrick-Schlein-Staffilani [23]. Together with an anisotropic version of the generalized lens transform in Carles [3], we derive rigorously the cubic NLS with anisotropic switchable quadratic traps in 2d through a modified Elgart-Erd\"os-Schlein-Yau procedure. For the 3d case, we establish the uniqueness of the corresponding Gross-Pitaevskii hierarchy without the assumption of factorized initial data.Comment: v6, 32 pages. Added an algebraic explanation of the generalized lens transform using the metaplectic representation. Accepted to appear in Journal de Math\'ematiques Pures et Appliqu\'ees. Comments are welcome

Second-order corrections to mean-field evolution of weakly interacting Bosons, II

We study the evolution of a N-body weakly interacting system of Bosons. Our work forms an extension of our previous paper I, in which we derived a second-order correction to a mean-field evolution law for coherent states in the presence of small interaction potential. Here, we remove the assumption of smallness of the interaction potential and prove global existence of solutions to the equation for the second-order correction. This implies an improved Fock-space estimate for our approximation of the N-body state

Nonconcentration of energy for a semilinear Skyrme model

We continue our investigation of a model introduced by Adkins and Nappi, in which omega mesons stabilize chiral solitons. The aim of this article is to show that the energy associated to equivariant solutions does not concentrate.Comment: 12 pages, 2 figure