We consider a random, uniformly elliptic coefficient field a(x) on the d-dimensional integer lattice Z d . We are interested in the spatial decay of the quenched elliptic Green function G(a; x, y). Next to stationarity, we assume that the spatial correlation of the coefficient field decays sufficiently fast to the effect that a logarithmic Sobolev inequality holds for the ensemble · . We prove that all stochastic moments of the first and second mixed derivatives of the Green function, that is, |∇xG(x, y)| p and |∇x∇yG(x, y)| p , have the same decay rates in |x − y| ≫ 1 as for the constant coefficient Green function, respectively. This result relies on and substantially extends the one by Delmotte and Deuschel [8], which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of G, that is, |∇xG(x, y)| 2 and |∇x∇yG(x, y)| . As an application, we are able to obtain optimal estimates on the random part of the homogenization error even for large ellipticity contrast. OutlineThe outline of this work is as follows: After introducing the discrete setting in Section 1, we present the statistical assumptions and the main result on the annealed moments of the Green function in Section 2. The following two sections contain applications of the main result: We present optimal estimates on the random part of the homogenization error in Section 3 and Section 4 contains an annealed Hölder-estimate in the spirit of De Giorgi. In Section 5 we explain our main assumption, a logarithmic Sobolev inequality, which in particular holds for all independent, identically distributed coefficient fields. Section 6 contains the main ingredients of the proof of the annealed Green function estimatesin particular we recall the result by Delmotte and Deuschel [8]. All proofs are postponed until Section 7.

We consider the Cauchy problem for (energy-subcritical) nonlinear Schrödinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superfluid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities without any restriction on the rotation frequency, generalizing earlier results given in [11,12]. Moreover, we find that the rotation term has a considerable influence in proving finite time blow-up in the focusing case.

Abstract. We propose a simple, efficient, and accurate numerical method for simulating the dynamics of rotating Bose-Einstein condensates (BECs) in a rotational frame with or without longrange dipole-dipole interaction (DDI). We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with an angular momentum rotation term and/or long-range DDI, state the twodimensional (2D) GPE obtained from the 3D GPE via dimension reduction under anisotropic external potential, and review some dynamical laws related to the 2D and 3D GPEs. By introducing a rotating Lagrangian coordinate system, the original GPEs are reformulated to GPEs without the angular momentum rotation, which is replaced by a time-dependent potential in the new coordinate system. We then cast the conserved quantities and dynamical laws in the new rotating Lagrangian coordinates. Based on the new formulation of the GPE for rotating BECs in the rotating Lagrangian coordinates, a time-splitting spectral method is presented for computing the dynamics of rotating BECs. The new numerical method is explicit, simple to implement, unconditionally stable, and very efficient in computation. It is spectral-order accurate in space and second-order accurate in time and conserves the mass on the discrete level. We compare our method with some representative methods in the literature to demonstrate its efficiency and accuracy. In addition, the numerical method is applied to test the dynamical laws of rotating BECs such as the dynamics of condensate width, angular momentum expectation, and center of mass, and to investigate numerically the dynamics and interaction of quantized vortex lattices in rotating BECs without or with the long-range DDI. 1. Introduction. Bose-Einstein condensation (BEC), first observed in 1995 [4,18,23], has provided a platform to study the macroscopic quantum world. Later, with the observation of quantized vortices [2,19,34,35,37,39,50], rotating BECs have been extensively studied in the laboratory. The occurrence of quantized vortices is a hallmark of the superfluid nature of BECs. In addition, condensation of bosonic atoms and molecules with significant dipole moments whose interaction is both nonlocal and anisotropic has recently been achieved experimentally in trapped 52 Cr and 164 Dy gases [1,22,27,32,33,36,48].At temperatures T much smaller than the critical temperature T c , the properties of BEC in a rotating frame with long-range dipole-dipole interaction (DDI) are well

Abstract.A mathematical framework for optimal bilinear control of nonlinear Schrödinger equations of Gross-Pitaevskii type arising in the description of Bose-Einstein condensates is presented. The obtained results generalize earlier efforts found in the literature in several aspects. In particular, the cost induced by the physical workload over the control process is taken into account rather than the often used L 2 -or H 1 -norms for the cost of the control action. Well-posedness of the problem and existence of an optimal control are proved. In addition, the first order optimality system is rigorously derived. Also a numerical solution method is proposed, which is based on a Newton-type iteration, and used to solve several coherent quantum control problems.